Skill Sheets

1

SkillSHEETanswers

SkillSHEET 1.1Adding and subtracting whole numbers

Addition of whole numbers

To add larger numbers, write them in columns according to place value and then add them.

Try these

1

Add these numbers, setting them out in columns as shown.

a

+

34

b

65

c

86

+

65

+

77

+

95

d

482

e

123

f

1418

+

517

+

89

+

2765

2

Add these numbers, setting them out in columns as shown.

a

419

b

68 069

c

1231 708 317 48 097

+

20 111 8 34

+

4 254

+

6 276

d

347

e

696

f

3992818 3 421 811 1489

692

+

63 044 2798180

+

8943

+

1000

Arrange these numbers in columns, then add them.1462 + 78 + 316

THINK WRITE

Set out the sum in columns. 1 4 6 27 8

+ 31116

1 8 5 6

Add the digits in the units column in your head (2 + 8 + 6 = 16). Write the 6 in the units column of your answer and carry the 1 to the tens column as shown in red.Now add the digits in the tens column (1 + 6 + 7 + 1 = 15). Write the 5 in the tens column of your answer and carry the 1 to the hundreds column as shown in orange.Add the digits in the hundreds column (1 + 4 + 3 = 8). Write 8 in the hundreds column of your answer as shown in green. There is nothing to carry.There is only a 1 in the thousands column. Write 1 in the thousands column of your answer.

12

3

4

5

1WORKEDExample

2

SkillSHEETanswers

3

Arrange these numbers in columns, and then add them.

a

137

+

841

b

723

+

432

c

149

+

562

+

55

d

47

+

198

+

12

e

376

+

948

+

11

f

8312

+

742

+

2693

Subtraction of whole numbers

For subtracting numbers, the digits are lined up vertically according to place value, as we saw for addition.The most commonly used method of subtraction is called the decomposition method. This is because thelarger number is decomposed or ‘taken apart’.

The 10 which is added to the top number is taken from the previous column of the same number.

So 32

−

14 is written as

32

−

14

and becomes

2

3

1

2

−

2

1

2

4

Now 4 can be taken from 12 and 10 from 20 to give 18.

2

3

1

2

−

1

1

4

1 8

3

SkillSHEETanswers

Try these

4

Evaluate:

a

98

−

54

b

167

−

132

c

47 836

−

12 713

d

149

−

63

e

642 803

−

58 204

f

3642

−

1811

g

664

−

397

h

12 900

− 8487 i 69 000 − 3561

Evaluate: a 6892 − 467 b 3000 − 467.

THINK WRITE

a Since 7 cannot be subtracted from 2, take one ten from the tens column of the larger number and add it to the units column of the same number. So the 2 becomes 12, and the 9 tens become 8 tens.

a 6 88912− 4 6 7

6 4 2 5

Subtract the 7 units from the 12 units (12 − 7 = 5).

Now subtract 6 tens from the 8 remaining tens (8 − 6 = 2).

Subtract 4 hundreds from the 8 hundreds (8 − 4 = 4).

Subtract 0 thousands from the 6 thousands (6 − 0 = 6).

b Since 7 cannot be taken from 0, 0 needs to become 10.

b −23909010−23949617

−22959313

We cannot take 10 from the tens column, as it is also 0. The first column that we can take anything from is the thousands, so 3000 is decomposed to 2 thousands, 9 hundreds, 9 tens and 10 units.

Now the subtraction will be straightforward. Subtract the units (10 − 7 = 3).

Subtract the tens (9 − 6 = 3).

Subtract the hundreds (9 − 4 = 5).

1

2

3

4

5

1

2

3

4

5

2WORKEDExample

4

SkillSHEETanswersSkillSHEET 1.2Multiplying whole numbersShort multiplication is used when we need to multiply by numbers less than or equal to 12. To multiply bynumbers greater than 12, long multiplication is used.

Calculate 735 ¥ 7 using short multiplication.

THINK WRITE

Set out the question. 273335× 75313435

Multiply 7 by the number in the units place: 7 × 5 = 35. Put 5 in the units place (of the answer part) and carry the 3 to the tens column.Multiply 7 by the number in the tens place: 7 × 3 = 21. Adding the ‘carry over’ gives 21 + 3 = 24. So put 4 in the tens place (of the answer part) and carry the 4 to the hundreds column.Multiply 7 by the number in the hundreds place: 7 × 7 = 49. Adding the ‘carry over’ gives 49 + 2 = 51. Write down 51 (in the answer part) so that 1 is in the hundreds and 5 is in the thousands column.

1

2

3

4

1WORKEDExample

Calculate 7623 ¥ 46 using long multiplication.

THINK WRITE

Set out the question one number under the other and underline. 37161213× 416

415171318Multiply 7623 by 6 (that is, by units) using the method described in worked example 1. Record your answer under the line.

Move to the next line. Put a 0 in the units column; multiply 7623 by 4 (that is, by tens) and record your answer next to 0.

27161213× 416

41517131831014191210

Underline and add the two parts together to find the final answer. 7161213× 416415171318

+ 3101419121031510161518

12

3

4

2WORKEDExample

5

SkillSHEETanswersTry these1 Calculate the following using short multiplication.

a × 16 3 b × 34 7 c × 25 8× 1 4 × 3 5 × 2 3

d × 76 e × 62 f × 428× 6 × 9 × 4

g × 563 h × 367 i × 845× 7 × 5 × 6

j × 956 k × 6750 l × 8253× 3 × 4 × 9

m × 7196 n × 4936 o × 5039× 8 × 3 × 5

2 Calculate the following using long multiplication.

a × 7 3 b × 4 9 c × 5 4× 1 6 × 2 4 × 3 7

× _ _ _ 8 × _ _ 9 _ × _ _ _ 8

+ _ _ 3 0 + _ _ _ 0 + _ _ 2 _

+ _ _ _ 8 + _ _ _ _ + _ _ _ _

d × 35 e × 68 f × 156× 18 × 59 × 29

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. . . . . . . . . .......... ..........

6

SkillSHEETanswersg × 641 h × 723 i × 824

× 36 × 17 × 53

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j × 395 k × 7462 l × 3758× 48 × 135 × 469

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m × 2183 n × 9241 o × 4569× 516 × 372 × 284

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SkillSHEETanswersSkillSHEET 1.3Rounding decimals to 2 decimal placesTo round a decimal correct to 2 decimal places, follow these steps:1. Consider the digit in the third decimal place (that is, the thousandth’s place).2. If it is less than 5, simply omit this digit and all digits that follow. (That is, omit all digits beginning from

the third decimal place.) 3. If it is 5 or greater than 5, add 1 to the preceding digit (that is, the one in the hundredth’s place) and omit

all digits beginning from the third decimal place.

Note: The sign ≈ is read as ‘is approximately equal to’.

Try theseRound each of the following numbers correct to 2 decimal places.

1 0.322 ≈ . . . . . . . . . . . . . . 2 0.257 ≈ . . . . . . . . . . . . . . 3 1.723 ≈ . . . . . . . . . . . . . . 4 2.555 ≈ . . . . . . . . . . . . . .

5 4.308 ≈ . . . . . . . . . . . . . . 6 12.195 ≈ . . . . . . . . . . . . . . 7 8.4678 ≈ . . . . . . . . . . . . . . 8 25.033 78 ≈ . . . . . . . . . . . .

9 18.333 333 ≈ . . . . . . . . . . . . . . 10 0.166 666 6 ≈ . . . . . . . . . . . . . .

Round each of the following numbers correct to 2 decimal places.a 0.239 b 4.5842

THINK WRITE

a The digit in the third decimal place is 9, which is greater than 5. So add 1 to the preceding digit (that is, to 3) and omit 9.

a 0.239 ≈ 0.24

b The digit in the third decimal place is 4. Since it is less than 5, simply omit all digits beginning from the third decimal place (that is, omit 4 and 2).

b 4.5842 ≈ 4.58

WORKEDExample

8

SkillSHEETanswersSkillSHEET 1.4Order of operations IIf an expression contains more than one operation, the calculations must be performed in the followingorder:1. brackets2. multiplication and division (from left to right)3. addition and subtraction (from left to right).

Try theseFind the value of each of the following, using the order of operations rules.

1 = 23 − (15 − 6) ÷ 3 2 = 40 − 8 × 4 + 2

= 23 − __ ÷ 3 = 40 − __ __ + 2

= 23 − __ = __ + 2

= __ __ = __ __

3 = 56 − 49 ÷ 7 4 = 12 ÷ (54 ÷ 9) + 3

= 56 − __ = 12 ÷ __ + __

= __ __ = __ + __

= __

Find the value of each of the following, using the order of operations rules.a 12 ∏ 2 + 4 ¥ 5b 15 + (6 + 4) ¥ 7 ∏ 10

THINK WRITE

a Write the question. a 12 ÷ 2 + 4 × 5There are three operations to be performed: division, addition and multiplication. According to the order of operations rules, multiplication and division must be done before the addition, from left to right. So, perform the division first, followed by the multiplication and finally the addition.

= 6 + 4 × 5= 6 + 20= 26

b Write the question. b 15 + (6 + 4) × 7 ÷ 10The calculation in the brackets must be done first, followed by multiplication and division. These are done in order of appearance (that is, from left to right), so in this case multiplication must be done before division. Addition is done last.

= 15 + 10 × 7 ÷ 10= 15 + 70 ÷ 10= 15 + 7= 22

12

12

WORKEDExample

9

SkillSHEETanswers5 = 14 − (16 − 5) ÷ 11 6 = 18 + 6 × 8 ÷ 8 × 3

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7 = 10 + 8 × 2 ÷ 8 8 = 144 ÷ 12 + 6 × 2 – 5

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9 = 10 + 6 × 3 ÷ 6 × 8 10 = 72 − 64 ÷ 8

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11 = 18 − 7 × 2 + 9 12 = 2 × (63 ÷ 9) + 3 × 2

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13 = 12 + 12 × 2 ÷ 6 14 = 21 ÷ (4 + 3) × 9

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10

SkillSHEETanswers15 = 6 + 6 × 4 + 8 16 = (6 + 3) × (7 + 2)

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17 = 11 × 8 + 9 × 3 18 = 7 + 3 × (4 + 3)

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19 = 8 + 4 × (6 + 3 × 2) 20 = (7 + 2) ÷ (12 − 3)

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11

SkillSHEETanswersSkillSHEET 1.5FactorsA number that divides exactly into another number is a factor of that number. The result of the division(that is, the quotient) is also a factor. These two factors form a factor pair. For example, if 7 is a factor of35, then 5 is also a factor of 35, since 35 ÷ 7 = 5, and so 5 and 7 form a pair of factors of 35.

We can find all the factors of a number by considering all the factor pairs of that number.

Note: There is an alternative way of finding a missing factor. For example, if 2 is a factor of 12, we canobtain the other factor by finding the number that will give 12 when multiplied by 2. That is, since 2 × 6 = 12,2 and 6 form a pair of factors of 12.

Try these1 a Find the missing factor in each of the following pairs of factors of 24:

3 and ____ , 6 and ____ , 12 and ____ .

b Find the missing factor in each of the following pairs of factors of 39:

3 and ____ , 39 and ____ .

c Find the missing factor in each of the following pairs of factors of 60:

4 and ____ , 5 and ____ , 10 and ____ , 20 and ____ , 30 and ____ .

d Find the missing factor in each of the following pairs of factors of 48:


Find the missing factor in each of the following factor pairs of 12:2 and __ , 4 and __ .

THINK WRITE

To find the missing factor, divide 12 by the known factor of the pair.

12 ÷ 2 = 6So a factor pair of 12 is 2 and 6.

Repeat step 1 for the second pair. 12 ÷ 4 = 3So a factor pair of 12 is 4 and 3.

1

2

1WORKEDExample

Find all the factors of 12.

THINK WRITE

Look for all the factor pairs of 12. Start with a number that divides exactly into 12.

Factor pairs of 12:1 and 122 and 63 and 4

List the factors of 12 in order from smallest to largest.

Factors of 12 are:1, 2, 3, 4, 6, 12.

1

2

2WORKEDExample

12

SkillSHEETanswers2 List all the factors of:

a 24 ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____

b 39 ____ , ____ , ____ , ____

c 60 ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____

d 48 ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____ , ____

3 For the following numbers, find all the factor pairs and then list all the factors.

a 8 b 50Factor pairs of 8: Factor pairs of 50:

1 and ____ ____ and ____

2 and ____ ____ and ____

____ and ____

Factors of 8 are Factors of 50 are

____ , ____ , ____ , ____ ____ , ____ , ____ , ____ , ____ , ____

c 24 d 45Factor pairs of 24: Factor pairs of 45:

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Factors of 24 are Factors of 45 are

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13

SkillSHEETanswersSkillSHEET 1.6MultiplesA multiple is the result of multiplication of one number by another number. For example, 10 is a multipleof both 2 and 5, since 2 × 5 = 10.

Try theseWrite the first five multiples of each of the following numbers.

1 3 The first five multiples of 3 are: ____, ____, ____, ____, ____










Write the first five multiples of 2.

THINK WRITE

To find the first five multiples of 2, multiply 2 by 1, then by 2, 3, 4 and finally by 5. Each product (that is, the result of each multiplication) is a multiple of 2.

2 × 1 = 22 × 2 = 42 × 3 = 62 × 4 = 82 × 5 = 10

List the first five multiples of 2. The first five multiples of 2 are: 2, 4, 6, 8, 10.

1

2

WORKEDExample

14

SkillSHEETanswersSkillSHEET 1.7Simplifying fractionsTo simplify a fraction, divide both numerator and denominator by the highest common factor (HCF).

Try theseWrite each of the following fractions in simplest form.

1 = 2 = 3 =

4 = 5 = 6 =

7 = 8 = 9 =

10 = 11 = 12 =

13 = 14 = 15 =

Write in simplest form.

THINK WRITE

The highest common factor of 8 and 52 is 4. So divide both numerator and denominator by 4.

=

852------

852------ 2

13------

WORKEDExample

1352------ 1

.......-------- 26

52------ .......

2-------- 4

52------ 1

.......--------

1252------ .......

.......-------- 48

52------ .......

.......-------- 2

6--- .......

.......--------

46--- .......

.......-------- 8

10------ .......

.......-------- 15

20------ .......

.......--------

1220------ .......

.......-------- 8

26------ .......

.......-------- 24

42------ .......

.......--------

1848------ .......

.......-------- 7

56------ .......

.......-------- 12

36------ .......

.......--------

15

SkillSHEETanswers

SkillSHEET 1.8Converting an improper fraction into a mixed number

To convert an improper fraction into a mixed number, divide the numerator by the denominator. The quo-tient will be the whole part of the mixed number, and the remainder will be the numerator of the fractionalpart. (The denominator of the fractional part is the same as the denominator of the improper fraction.)

Try these

Change each of the following improper fractions into a mixed number.

1

=

1

2

=

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3

=

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4

=

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5

=

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6

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7

=

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8

=

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9

=

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10

=

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Change into a mixed number.

THINK WRITE

Divide 7 by 5. Five goes into seven once with a remainder of 2. So the whole part of the mixed number is 1 and the numerator of the fractional part is 2. (The denominator remains unchanged.)

= 1

75---

75--- 2

5---

WORKEDExample

95--- .......

5-------- 7

3--- 1

3--- 8

3---

175

------ 127

------ 143

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174

------ 236

------ 425

------

358

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16

SkillSHEETanswersSkillSHEET 1.9Finding and converting to the lowest common denominatorTo obtain the lowest common denominator (LCD) of two or more fractions, we need to find the lowestcommon multiple (LCM) of the denominators of these fractions.

To convert a fraction to the LCD, we need to establish how many times the original denominator fits ordivides into the LCD. We then need to multiply both numerator and denominator of the fraction by thatnumber.

Find the lowest common denominator of and .

THINK WRITE

To obtain the lowest common denominator of the given fractions, we need to find the lowest common multiple (LCM) of 6 and 8. To find the LCM, list some multiples of 6 and 8 and select the smallest number that is on both lists.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64LCM = 24

The LCM represents the lowest common denominator, so write the answer.

LCD of and is 24.

56--- 3

8---

1

256--- 3

8---

1WORKEDExample

Convert and to fractions with the lowest common denominator of 24.

THINK WRITE

Consider . The denominator 6 goes into 24

(the LCD) 4 times. So multiply both

numerator and denominator of by 4.

= =

Consider . The denominator 8 goes into 24

(the LCD) 3 times. So multiply both

numerator and denominator of by 3.

= =

56--- 3

8---

156---

56---

56--- 5 4×

6 4×------------ 20

24------

238---

38---

38--- 3 3×

8 3×------------ 9

24------

2WORKEDExample

17

SkillSHEETanswersTry these1 Find the lowest common denominator of each of the following pair of fractions.

a and b and

Multiples of 7: 7, 14, 21, . . . . . . . . . . . . . Multiples of 4: 4, 8, 12, . . . . . . . . . . . . .

Multiples of 6: 6, 12, 18, . . . . . . . . . . . . . Multiples of 3: 3, 6, 9, . . . . . . . . . . . . .

LCM = . . . . . . . . . . . . . LCM = . . . . . . . . . . . . .

c and d and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

e and f and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g and h and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i and j and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Convert each pair of fractions in question 1 to fractions with their respective lowest commondenominators.

a = = . . . . . . . . . . . . . . . . . . . . . . . . . . . . b = = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= = . . . . . . . . . . . . . . . . . . . . . . . . . . . . = = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c = . . . . . . . . . . . . . . . . . . . . . . . . . . . . d = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57--- 1

6--- 3

4--- 2

3---

16--- 1

4--- 2

3--- 3

5---

23--- 5

6--- 1

6--- 5

8---

38--- 3

4--- 5

9--- 5

6---

13--- 4

9--- 5

12------ 7

8---

57--- 5 6×

7 6×------------ 3

4--- ....... .......×

....... .......×-----------------------

16--- 1 7×

6 7×------------ 2

3--- ....... .......×

....... .......×-----------------------

16--- 2

3---

14--- 3

5---

18

SkillSHEETanswerse = . . . . . . . . . . . . . . . . . . . . . . . . . . . . f = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g = . . . . . . . . . . . . . . . . . . . . . . . . . . . . h = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i = . . . . . . . . . . . . . . . . . . . . . . . . . . . . j = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23--- 1

6---

56--- 5

8---

38--- 5

9---

34--- 5

6---

13--- 5

12------

49--- 7

8---

19

SkillSHEETanswersSkillSHEET 1.10Adding and subtracting fractions IAdding and subtracting fractions with the same denominatorTo add (or subtract) fractions with the same denominator, add (or subtract) numerators and leave thedenominator unchanged.

Try these1 Evaluate each of the following.

a + b +

= =

= =

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= 1

Evaluate each of the following.

a + b 1 -

THINK WRITE

a Add numerators together and leave the denominator unchanged.

a + =

=

Simplify by dividing the numerator and the denominator by 4.

=

Convert the improper fraction into a mixed numeral.

= 1

b Change the mixed number to an improper fraction.

b 1 − = −

Subtract the numerators (subtract 3 from 7) and leave the denominator unchanged.

=

=

78--- 5

8--- 2

5--- 3

5---

178--- 5

8--- 7 5+

8------------

128

------

232---

312---

125--- 3

5--- 7

5--- 3

5---

27 3–

5------------

45---

1WORKEDExample

14--- 3

4--- 2

5--- 4

5---

1 .......+4

----------------- ....... .......+5

-----------------------

.......

4-------- .......

5--------

.......

5--------

20

SkillSHEETanswersc + d −

= =

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

e − f 1 −

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= −

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

=

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g 1 − h 1 −

= − = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i 3 − j 4 −

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59--- 2

9--- 8

11------ 5

11------

....... .......+

.......----------------------- ....... .......+

.......-----------------------

78--- 3

8--- 1

7--- 3

7---

87--- .......

7--------

8 .......–7

-----------------

38--- 5

8--- 2

5--- 4

5---

.......

8-------- .......

.......--------

15--- 3

5--- 3

10------ 7

10------

21

SkillSHEETanswersAdding and subtracting fractions when one denominator is a multiple of the otherTo add (or subtract) two fractions when one denominator is a multiple of the other, follow these steps.

Step 1 Keep the fraction with the larger denominator the same.

Step 2 Work out what number needs to be multiplied to the denominator of the other fraction to obtain thevalue of the larger denominator.

Step 3 Multiply both the numerator and denominator of this fraction by the number found in step 2 toobtain an equivalent fraction.

Step 4 Add (or subtract) the numerators together and leave the denominator unchanged.

Step 5 Simplify (if necessary); if the result is an improper fraction, convert it to a mixed number.

Evaluate:

a − b 2 +

THINK WRITE

a Notice that 6 is a multiple of 3, so keep

the same and multiply both the numerator

and the denominator of by 2.

a − = − ×

= −

As both fractions now have the same denominator, subtract the numerators and leave the denominator unchanged.

=

=

Write the fraction in simplest form by dividing both the numerator and the denominator by 3.

=

b Change the mixed number to an improper fraction.

b 2 + = +

As 8 is a multiple of 2, keep the same

and convert to an equivalent fraction

with a denominator of 8. (Multiply both the numerator and the denominator by 4.)

= + ×

= +

As both fractions now have the same denominator, add the numerators and leave the denominator unchanged.

=

=

Change the improper fraction to a mixed number.

= 2

56--- 1

3--- 1

8--- 1

2---

156---

13---

56--- 1

3--- 5

6--- 1

3--- 2

2---

56--- 2

6---

25 2–

6------------

36---

312---

118--- 1

2--- 17

8------ 1

2---

2178

------

12---

178

------ 12--- 4

4---

178

------ 48---

317 4+

8---------------

218

------

458---

2WORKEDExample

22

SkillSHEETanswersTry these2 Evaluate each of the following.

a − b + c +

= × − = + × = × +

= − = + = +

= = =

= = =

= 1

d − e 1 + f 3 −

= − × = + = −

= − = × + = − ×

= = + = −

= = =

= =

= . . . . . .

= . . . . . .

25--- 1

10------ 7

12------ 5

6--- 2

3--- 5

18------

25--- .......

.......-------- 1

10------ .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- 1

10------ .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

....... .......–10

----------------------- ....... .......+12

----------------------- ....... .......+

.......-----------------------

.......

10-------- .......

.......-------- .......

.......--------

.......

.......--------

1120------ 2

5--- 3

4--- 7

12------ 2

9--- 2

3---

.......

.......-------- .......

.......-------- .......

.......-------- .......

4-------- 7

12------ .......

9-------- 2

3---

.......

.......-------- .......

20-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

....... .......–

.......----------------------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- ....... .......+

.......----------------------- ....... .......–

.......-----------------------

.......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

23

SkillSHEETanswersSkillSHEET 1.11Adding and subtracting fractions IITo add or subtract fractions, follow these steps.Step 1 Convert all mixed numbers to improper fractions first.Step 2 Convert (if necessary) to equivalent fractions with a common denominator.Step 3 Add (subtract) numerators together and leave the denominator unchanged.Step 4 Simplify (if necessary); if the result is an improper fraction, convert it to a mixed number.

Calculate each of the following.

a + b − c 1 + 2

THINK WRITE

a The LCD of 5 and 8 is 40. To obtain equivalent fractions with a denominator of 40, multiply the numerator and denominator of the first fraction by 8 and the second fraction by 5.

a + = × + ×

= +

Add numerators (16 and 35) together and leave the denominator (40) unchanged.

=

Convert the improper fraction to a mixed number.

= 1

b The LCD of 4 and 6 is 12. To obtain equivalent fractions with a denominator of 12, multiply the numerator and denominator of the first fraction by 3 and the second fraction by 2.

b − = × − ×

= −

Subtract the numerators (9 − 2) and leave the denominator (12) unchanged.

=

c Convert mixed numbers to improper fractions.

c 1 + 2 = +

The LCD of 2 and 8 is 8. To obtain an equivalent fraction with a denominator of 8, multiply the numerator and denominator of the first fraction by 4. (The denominator of the second fraction is already 8, so this fraction does not need to be changed.)

= × +

= +


=


= 4

25--- 7

8--- 3

4--- 1

6--- 1

2--- 5

8---

125--- 7

8--- 2

5--- 8

8--- 7

8--- 5

5---

1640------ 35

40------

25140------

31140------

134--- 1

6--- 3

4--- 3

3--- 1

6--- 2

2---

912------ 2

12------

27

12------

112--- 5

8--- 3

2--- 21

8------

232--- 4

4--- 21

8------

128

------ 218

------

3338

------

418---

WORKEDExample

24

SkillSHEETanswersTry these1 Calculate each of the following.

a + b − c 1 + 1

= × + × = − × = +

= + = − = × + ×

= = = +

=

= . . . . . . . . . . . . . . . . . . . . . . . .

d + e − f 2 − 1

= × + × = × − × = −

= . . . . . . . . . . . .

+ . . . . . . . . . . . .

= . . . . . . . . . . . .

− . . . . . . . . . . . .

= × − ×

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . .

− . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

g + h − i + 2

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

.

15--- 3

8--- 5

8--- 1

2--- 2

3--- 1

8---

15--- .. .

8---- 3

8--- .. .

5---- 5

8--- 1

2--- 4

...---- 5

3--- .. .

8----

.. .

40------ .. .

40------ .. .

8---- .......

.......-------- 5

3--- .......

.......-------- 9

8--- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......--------

34--- 4

5--- 3

4--- 1

3--- 1

5--- 1

2---

34--- .......

.......-------- 4

5--- .......

.......-------- .......

.......-------- .......

.......-------- .. .

3---- .......

.......-------- 11

...------ .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

12--- 1

3--- 4

5--- 2

7--- 2

9--- 1

3---

SkillSHEETanswers

25

SkillSHEETanswersj + k − l 3 − 1

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

710------ 7

8--- 5

6--- 1

8--- 1

4--- 1

3---

26

SkillSHEETanswersSkillSHEET 1.12Converting a mixed number into an improper fractionTo convert a mixed number into an improper fraction, follow these steps.1. Multiply the denominator by the whole part and add the numerator. 2. Put the resultant number as the numerator of the improper fraction. 3. Write the denominator. (It is the same as the one in the mixed number.)

Try theseConvert each of the following mixed numbers into an improper fraction.

1 1 = 2 2 = 3 1 =

4 2 = . . . . . . . . . . . . . . . . . . . . . . .

5 3 = . . . . . . . . . . . . . . . . . . . . . .

6 2 = . . . . . . . . . . . . . . . . . . . . . .

7 4 = . . . . . . . . . . . . . . . . . . . . . .

8 1 = . . . . . . . . . . . . . . . . . . . . . .

9 5 = . . . . . . . . . . . . . . . . . . . . . .

10 4 = . . . . . . . . . . . . . . . . . . . . . .

Convert 2 into an improper fraction.

THINK WRITE

To obtain the numerator of the improper fraction, multiply the denominator by the whole part and add the numerator. (That is, the numerator is 8 × 2 + 3 = 19.) The denominator of the improper fraction is the same as the one in the mixed number.

2 =

38---

38--- 19

8------

WORKEDExample

35---

1 5× .......+5

--------------------------- 58---

2 .......× 5+8

--------------------------- 27---

....... .......× .......+.......

--------------------------------------

78--- 1

2--- 4

5---

34--- 1

9--- 5

6---

37---

27

SkillSHEETanswersSkillSHEET 1.13Multiplying and dividing fractionsMultiplying fractionsTo multiply fractions follow these steps.Step 1 Convert all mixed numbers into improper fractions.Step 2 Simplify as much as possible.Step 3 Multiply numerators together and multiply denominators together.Step 4 If the answer is an improper fraction, convert it to a mixed number.

Perform the following multiplications.

a ¥ b 1 ¥

THINK WRITE

a Write the multiplication problem. a

Cross-cancel 3 and 6 by dividing each by 3 (that is, 3 ÷ 3 = 1; 6 ÷ 3 = 2).

=

Multiply the numerators together and denominators together. =

b Write the multiplication problem. b 1

Convert the mixed number into an improper fraction. =

Cross-cancel 7 and 7 by dividing each by 7 (that is, 7 ÷ 7 = 1). Next cross-cancel 6 and 4 by dividing each by 2 (that is, 4 ÷ 2 = 2; 6 ÷ 2 = 3).

=


Convert an improper fraction into a mixed number. = 1

35--- 1

6--- 3

4--- 6

7---

135--- 1

6---×

215--- 1

2---×

31

10------

134--- × 6

7---

274--- 6

7---×

312--- 3

1---×

432---

512---

WORKEDExample 1

28

SkillSHEETanswersTry these1 Perform each of the following multiplications.

a × b × c ×

= × = = ×

= =

d × e 1 × f ×

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

g × 2 h × i 1 × 2

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

25--- 1

2--- 1

3--- 1

2--- 3

10------ 1

3---

15--- .......

.......-------- .. .

6---- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

34--- 1

2--- 2

3--- 1

2--- 7

8--- 10

3------

45--- 1

4--- 4

9--- 18

4------ 1

8--- 2

5---

29

SkillSHEETanswersDividing fractionsTo divide fractions follow these steps.1. Convert mixed numbers to improper fractions.2. Turn the second fraction upside down; replace ÷ with × and perform multiplication (tip and turn).3. If the result is an improper fraction, convert it to a mixed number.

Try these2 Calculate each of the following.

a ÷ b ÷ c ÷

= × = × = ×

= × = = ×

= =


a ÷ b 1 ÷

THINK WRITE

a Turn the second fraction upside down and replace ÷ with ×. a =

Multiply numerators together and denominators together. =

Convert the improper fraction into a mixed number. = 1

b Convert the mixed number into an improper fraction first. b 1 =

Turn the second fraction upside down and replace ÷ with ×. =

Cross-cancel 4 and 2 by dividing each by 2 (i.e. 4 ÷ 2 = 2; 2 ÷ 2 = 1).

=



38--- 1

3--- 1

4--- 1

2---

138--- 1

3---÷ 3

8--- 3

1---×

298---

318---

114--- 1

2---÷ 5

4--- 1

2---÷

254--- 2

1---×

3 52--- 1

1---×

452---

512---

WORKEDExample 2

38--- 3

4--- 3

5--- 5

6--- 1

8--- 1

2---

38--- .. .

3---- 3

5--- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

30

SkillSHEETanswersd ÷ e 1 ÷ f ÷ 1

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

g 1 ÷ h ÷ 1 i 2 ÷

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

49--- 1

3--- 1

4--- 3

8--- 1

4--- 1

2---

35--- 4

5--- 2

3--- 5

7--- 3

4--- 1

6---

31

SkillSHEETanswersSkillSHEET 1.14Adding and subtracting decimalsAdding decimals with the same number of decimal placesTo add decimals with the same number of decimal places, follow these steps.Step 1 Set out the addition as you would for whole numbers. (As a result, decimal points will be in line

and digits of the same place value will be underneath each other.)Step 2 Moving from right to left, add numbers together as if they were whole numbers.Step 3 Put the decimal point in the answer directly underneath the decimal points in the question.

Try these1 Rewrite the following in columns and then add.

a 2.3 + 4.2 b 7.1 + 1.4 c 2.31 + 5.24 d 0.25 + 0.76 e 3.54 + 4.79

2 . __ __ . __ 2 . __ __ __ . __ __ + . . . . . . . . . . . . . .

+ 4 . 2 + 1 . 4 + 5 . 2 4 + 0 . __ __ + . . . . . . . . . . . . . .

__ . 5 __ . __ __ . __ __ __ . __ __ + . . . . . . . . . . . . . .

f 19.27 + 32.55 g 38.13 + 12.08 h 9.155 + 6.407

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

i 8.0013 + 1.1572 j 9.7216 + 3.4201

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

Rewrite the following in columns and then add.2.35 + 4.52

THINK WRITE

Write the numbers under each other as you would if they were whole; then underline. Add the digits in columns: 5 + 2 = 7; 3 + 5 = 8; 2 + 4 = 6. Put the decimal point in the answer directly underneath the decimal points in the question (that is, between 6 and 8).

+ 2.35+ 4.52

+ 6.87

1WORKEDExample

SkillSHEETanswers

32

SkillSHEETanswersAdding decimals with a different number of decimal placesTo add decimals with a different number of decimal places, follow these steps.Step 1 Rewrite the numbers in columns, positioning them so that the decimal points are in line and digits

of the same place value are underneath each other.Step 2 Add zeros where necessary to make the numbers equal in ‘length’ (i.e. to have the same number of

digits after the decimal point).Step 3 Moving from right to left, add numbers together as if they were whole numbers.Step 4 Put the decimal point in the answer directly under the decimal points in the question.

Try these2 Rewrite the following in columns and then add.

a 2.318 + 4.2 b 7.156 + 1.4 c 2.31 + 5.242 d 0.9 + 0.76

2 . 3 __ __ __ . __ 5 6 __ . __ __ 0 __ . __ 0

+ 4 . 2 0 0 + 1 . 4 0 0 + __ . 2 4 2 + __ . __ __

__ . __ 1 8 __ . __ __ __ __ . __ __ __ __ . __ __

e 3.74 + 4.791 f 19.27 + 12.5 g 28.103 + 12.08 h 7.155 + 3.07

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

i 8.3 + 1.157 j 9.72 + 4.8201

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . .

Rewrite the following in columns and then add.2.3 + 4.178

THINK WRITE

Position the two numbers so that the decimal points are in line. Add two zeros to the first number so that both numbers contain 3 digits after the decimal point. Add the numbers in columns (add 0 to 8, 0 to 7, 3 to 1 and 2 to 4). The decimal point in the answer must be directly below the decimal points in the question (that is, between 6 and 4).

+ 2.300+ 4.178

+ 6.478

2WORKEDExample

33

SkillSHEETanswersSubtracting decimals with the same number of decimal placesTo subtract decimals with the same number of decimal places, follow these steps.Step 1 Set out the subtraction as you would for whole numbers. (As a result, decimal points will be in line

and digits of the same place value will be underneath each other.)Step 2 Moving from right to left, subtract numbers in columns as if they were whole numbers.Step 3 Put the decimal point in the answer directly underneath the decimal points in the question.

Try these3 Rewrite the following in columns and then subtract.

a 7.3 − 4.2 b 7.1 − 1.4 c 5.31 − 3.24 d 0.85 − 0.76

7 . __ __ . __ 5 . __ __ __ . __ __

− 4 . 2 − 1 . 4 − __ . 2 4 − 0 . __ __

__ . 1 __ . __ __ . __ __ __ . __ __

e 7.54 − 4.79 f 19.27 − 12.55 g 18.13 − 12.08 h 12.155 − 10.407

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

− . . . . . . . . . . . . . . − . . . . . . . . . . . . . . − . . . . . . . . . . . . . . − . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

i 8.0013 − 1.1572 j 9.7216 − 3.4201

+ . . . . . . . . . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . .

− . . . . . . . . . . . . . . . . . . . . . . − . . . . . . . . . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . .

Rewrite the following in columns and then subtract.6.35 - 4.52

THINK WRITE

Write the numbers under each other as you would if they were whole, then underline. Subtract the digits in columns. First subtract hundredths: 5 − 2 = 3. Next subtract tenths: since we cannot subtract 5 from 3, we need to borrow from 6. Cross out 6 and write 5 next to it; 3 becomes 13. Now subtract 5 from 13 (13 − 5 = 8). Finally subtract units: 5 − 4 = 1. Put the decimal point in the answer directly underneath the decimal points in the question (that is, between 1 and 8).

− 565.1315− 141.1512

− 111.1813

3WORKEDExample

34

SkillSHEETanswersSubtracting decimals with a different number of decimal placesTo subtract decimals with a different number of decimal places, follow these steps.Step 1 Rewrite the numbers in columns, positioning them so that the decimal points are in line and digits

of the same place value are underneath each other.Step 2 Add zeros where necessary to make the numbers equal in ‘length’ (that is, to have the same

number of digits after the decimal point).Step 3 Moving from right to left, subtract numbers in columns as you would if they were whole numbers.Step 4 Put the decimal point in the answer directly under the decimal points in the question.

Try these4 Rewrite the following in columns and then subtract.

a 5.318 − 4.2 b 7.156 − 1.4 c 8.31 − 5.242 d 0.9 − 0.76

5 . 3 __ __ __ . __ 5 6 __ . __ __ 0 __ . __ 0

− 4 . 2 0 0 − 1 . 4 0 0 − __ . 2 4 2 − __ . __ __

__ . __ 1 8 __ . __ __ __ __ . __ __ __ __ . __ __

e 13.74 − 4.791 f 19.27 − 12.5 g 28.103 − 12.08 h 7.155 − 3.07

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

− . . . . . . . . . . . . . . − . . . . . . . . . . . . . . − . . . . . . . . . . . . . . − . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . . + . . . . . . . . . . . . . .

i 8.3 − 1.157 j 9.72 − 4.8201

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . .

− . . . . . . . . . . . . . . − . . . . . . . . . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . .

Rewrite the following in columns and then subtract.2.3 - 0.17

THINK WRITE

Position the two numbers so that the decimal points are in line. Add one zero to the first number so that both numbers contain 2 digits after the dot. Start subtracting in columns. We cannot subtract 7 from 0, so we need to borrow from 2. Cross out 2 and write 1; 0 becomes 10. Now subtract: 10 − 7 = 3, 2 − 1 = 1 and 2 − 0 = 0. The decimal point in the answer must be directly below the decimal points in the question (that is, between 2 and 1).

− 21.2310− 01.1117

− 21.1113

4WORKEDExample

35

SkillSHEETanswersSkillSHEET 1.15Multiplying and dividing decimalsMultiplying decimalsTo multiply a decimal by a whole number, or by another decimal, first perform multiplication, ignoring thedecimal point(s), as you would if both were whole numbers. To find the position of the decimal point in theanswer, count the total number of decimal places in the numbers being multiplied. The number obtained forthe answer will have the same number of decimal places as the total number of decimal places in thenumbers being multiplied.

Note that when setting out multiplication, it is not necessary to position the numbers so that the decimalpoints are in line.

Try these1 Multiply each of the following.

a 7.9 × 6 b 6.5 × 5 c 1.79 × 7 d 6.2 × 0.5

57.9 6.5 1.79 . . . . . . . . . . . .

× 6 × 5 × 7 × . . . . . . . . . . . . .

__.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

e 9.6 f 1.85 g 7.3 h 7.3× 16 × 21 × 13 × 1.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiply each of the following.a 9.78 ¥ 5 b 1.31 ¥ 3.2

THINK WRITE

a Multiply, ignoring the decimal point (that is, multiply 978 by 5). Count the numbers after the decimal point — there are two (7 and 8). Put the decimal point in the answer so that there are two numbers after it.

a 394.4748× 54484.4940

b Multiply, ignoring the decimal point (that is, multiply 131 by 32). Count all numbers after the decimal points — there are 3 (two in the first number and one in the second). Put the decimal point in the answer so that it has three decimal places.

b 1.3.1× 3.2

2.6.23.9.3.04.1.9.2

WORKEDExample 1

36

SkillSHEETanswersi 8.9 j 9.42 k 64.5 l 5.72

× 3.5 × 1.2 × 5.2 × 9.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dividing decimals by a whole numberTo divide a decimal by a whole number, follow these steps.

1. Set out the question as you would for division of whole numbers.

2. Divide as for whole numbers.

3. Put the decimal point in the answer: it has to be directly above the decimal point in the question.

Calculate each of the following.a 3.06 ∏ 3b 10.024 ∏ 4

THINK WRITE

a Set out the division. How many times does 3 go into 3? It goes once with no remainder, so put 1 directly above the 3 in the answer section. How many times does 3 go into 0? It goes 0 times, so put 0 above 0. The next number is 6, and 3 goes into 6 exactly twice, so put 2 above the 6. Now place the decimal point directly above the one in the question (that is, between 1 and 0).

a 1.023)3.06

3.06 ÷ 3 = 1.02

b The number 4 does not go into 1, so try 10. How many times does 4 go into 10? It goes twice with a remainder of 2. So put 2 above 10 and a small 2 (the remainder) in front of the next digit (which is 0). Now 2 and 0 together form a 20. How many times does 4 go into 20? It goes exactly 5 times, so put 5 above 20. Move to the next digit, which is 2. Since 4 does not go into 2, put 0 above 2 and carry the remainder in front of the next digit, which is 4. The remainder of 2 together with a 4 form 24; 4 goes into 24 exactly 6 times, so put 6 above 24. To complete the division, put the decimal point directly above the one in the question (that is, between 2 and 5).

b 2.50264)1020224

10.024 ÷ 4 = 2.506

WORKEDExample

.

2

37


a 2.46 ÷ 2 b 4.08 ÷ 2 c 3.60 ÷ 3 d 63.9 ÷ 3

2)1._ _ 2)_.0_ 2)_._ _ 2). . . . . . . . . . . . . .

2)2.4 6 2)4.0 8 3)3.6 0 3)6 3.9

2.46 ÷ 2 = ............ 4.08 ÷ 2 = ............ 3.60 ÷ 3 = ............ 63.9 ÷ 3 = ............

e 40.08 ÷ 4 f 50.5 ÷ 5 g 123.6 ÷ 6 h 5.764 ÷ 4

2). . . . . . . . . . . . . . 2). . . . . . . . . . . . . . 2). . . . . . . . . . . . . . 2). . . . . . . . . . . . . .

4)4 0.0 8 5)5 0.5 6)1 2 3.6 4)5.7 6 4

40.08 ÷ 4 = ............ 50.5 ÷ 5 = ............ 123.6 ÷ 6 = ............ 5.764 ÷ 4 = ............

i 5.095 ÷ 5 j 27.816 ÷ 6 k 687.4 ÷ 7 l 9.024 ÷ 8

2). . . . . . . . . . . . . . 2). . . . . . . . . . . . . . 2). . . . . . . . . . . . . . 2). . . . . . . . . . . . . .

5)5.0 9 5 6)2 7.8 1 6 7)6 8 7.4 8)9.0 2 4

5.095 ÷ 5 = ............ 27.816 ÷ 6 = ............ 687.4 ÷ 7 = ............ 9.024 ÷ 8 = ............

m 54.009 ÷ 9 n 75.812 ÷ 4 o 1003.2 ÷ 11 p 300.45 ÷ 5

2). . . . . . . . . . . . . . 2). . . . . . . . . . . . . . 22). . . . . . . . . . . . . . 2). . . . . . . . . . . . . .

9)5 4.0 0 9 4)7 5.8 1 2 11)1 0 0 3.2 5)3 0 0.4 5

54.009 ÷ 9 = ............ 75.812 ÷ 4 = ............ 1003.2 ÷ 11 = ............ 300.45 ÷ 5 = ............

38

SkillSHEETanswersSkillSHEET 1.16Dividing a decimal by a decimalTo divide decimals, we need to change the second decimal (that is, the divisor) into a whole number. Thiscan be done by moving the decimal point to the right until it is at the end of the number. The first number(that is, the dividend) needs to be adjusted accordingly. That is, we need to move the decimal point in thedividend to the right as many places as we did in the divisor. When these adjustments are done, we canproceed with the division. Divide as you would for whole numbers and place the decimal point in theanswer directly above the decimal point in the question.

Try theseCalculate each of the following.

1 9.236 ÷ 0.4 2 4.528 ÷ 0.2 3 52.38 ÷ 0.3 4 489.15 ÷ 0.5

= 92.36 ÷ 04. = 45.28 ÷ . . . . . . . . . . . . . = . . . . . . . . . . . . ÷ . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= 92.36 ÷ 4 = 45.28 ÷ . . . . . . . . . . . . . = . . . . . . . . . . . . ÷ . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

4)_ _ . _ _ . . . . . .)_ _ . _ _ . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

4)9 2 . 3 6 . . . . . .)4 5 . 2 8 . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

5 6.759 ÷ 0.02 6 5.2563 ÷ 0.03 7 24.928 ÷ 0.08 8 73.21 ÷ 0.04

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

Calculate 4.598 ∏ 0.02.

THINK WRITE

To make the divisor (0.02) a whole number, move the decimal point 2 places to the right so that it is at the end of the number. Also move the decimal point in the dividend 2 places to the right.

4.598 ÷ 0.02 = 459.8 ÷ 002.

Omit the zeros in front of the divisor and the decimal point at the end of it (that is, write 2 instead of 002.).

4.598 ÷ 0.23 = 459.8 ÷ 2

Set out the division. Divide as if both numbers are whole numbers. Place the decimal point in the answer directly above the decimal point in the question (that is, between 9 and 9).

2)212191.192)415191.18

1

2

3

WORKEDExample

39

SkillSHEETanswers9 4.581 ÷ 0.06 10 213.4 ÷ 0.04 11 171.6 ÷ 0.06 12 23.1 ÷ 0.11

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

13 10.056 ÷ 0.003 14 0.0004 ÷ 0.002 15 0.0255 ÷ 0.006 16 2.75 ÷ 0.005

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

40

SkillSHEETanswersSkillSHEET 1.17Rounding to the first (leading) digitTo round to the first (or leading) digit, follow these rules.1. Consider the digit after the leading one (i.e. the second digit). 2. If it is less than 5, leave the first digit unchanged and replace all other digits with zeros.3. If the second digit is 5 or greater than 5, add 1 to the leading digit and replace the rest of the digits with

zeros.

Note: The sign ≈ is read as ‘is approximately equal to’.

Try theseRound each of the following numbers to the first (or leading) digit.

1 84 ≈ 8__ 2 77 ≈ __0

3 120 ≈ __00 4 569 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 903 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7235 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 5502 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2970 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 23 004 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 59 455 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Round each of the following numbers to the first (or leading) digit.a 2371 b 872

THINK WRITE

a Since the second digit (3) is less than 5, replace all digits except the first one with zeros.

a 2371 ≈ 2000

b The second digit (7) is greater than 5. So add 1 to the leading digit and replace the rest of the digits with zeros.

b 872 ≈ 900

WORKEDExample

41

SkillSHEETanswersSkillSHEET 2.1Using < or > to compare the size of numbersThe sign < is read as ‘is less than’, and the sign > is read as ‘is greater than’. One way to remember whichsign is which is to remember that the open end of each sign faces the larger number in the pair, while theclosed end faces the smaller number.

Try theseInsert the inequality signs < or > to make each of the following statements true.

1 5 . . . . . . . . . . . . . . 12 2 7 . . . . . . . . . . . . . . 0 3 12 . . . . . . . . . . . . . . 21 4 77 . . . . . . . . . . . . . . 70

5 99 . . . . . . . . . . . . . . 19 6 20 . . . . . . . . . . . . . . 54 7 100 . . . . . . . . . . . . . . 87 8 56 . . . . . . . . . . . . . . 124

9 1000 . . . . . . . . . . . . . . . 110 10 850 . . . . . . . . . . . . . . 846

Insert the inequality signs < or > to make each of the following statements true.a 10 . . . . . . . 7 b 70 . . . . . . . 95

THINK WRITE

a 10 is larger than 7, so insert an ‘is greater than’ sign (>) between the two numbers.

a 10 > 7

b 70 is smaller than 95, so insert an ‘is less than’ sign (<) between the numbers.

b 70 < 95

WORKEDExample

42

SkillSHEETanswersSkillSHEET 2.2Ascending and descending orderTo arrange a set of numbers in ascending order, write them in order from smallest to largest.To arrange a set of numbers in descending order, write them in order from largest to smallest.

Try these1 Arrange each of the following sets of numbers in ascending order.

a 21, 12, 4, 19, 14 b 98, 89, 88, 92, 84

4, 12, . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . 84, . . . . . . . . . . . . , . . . . . . . . . . . . , 92, . . . . . . . . . . . .

c 20, 5, 44, 0, 13 d 54, 48, 76, 12, 60

. . . . . . . . . . . . , . . . . . . . . . . . . , 13, . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . .

2 Arrange each of the following sets of numbers in descending order.

a 23, 2, 14, 29, 24 b 9, 16, 48, 12, 34

29, 24, . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . 48, . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . , 9

c 32, 27, 40, 86, 69 d 106, 78, 116, 92, 54

. . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . , 32, . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . .

a Arrange the following set of numbers in ascending order: 7, 22, 0, 3, 12b Arrange the following set of numbers in descending order: 26, 15, 38, 9, 20

THINK WRITE

a ‘Ascending order’ means from smallest to largest. The smallest number in the given set is 0, so put it first. The second smallest number is 3, so write it next. Then follow with 7, 12 and 22 (with 22 being the largest).

a 0, 3, 7, 12, 22

b ‘Descending order’ means from largest to smallest. Select the largest number in the set (it is 38) and write it first. The second largest number is 26, so put it next. The next largest number is 20, followed by 15 and 9 (with 9 being the smallest).

b 38, 26, 20, 15, 9

WORKEDExample

43

SkillSHEETanswersSkillSHEET 2.3Marking numbers on a number lineNumber lines can be used to show various numbers. An individual number can be shown by placing a dotdirectly above the corresponding number on the number line.

When drawing a number line, make sure all intervals are of equal length.

Try these1 Draw a number line with 10 equal intervals marked from 0 to 10. Mark the following numbers on the

number line with a dot.

a 0 b 4 c 6 d 9

2 Draw a number line with 10 equal intervals marked from 10 to 20. Mark the following numbers on thenumber line with a dot.

a 11 b 15 c 18 d 20

3 Draw a number line with 10 equal intervals marked from 50 to 60. Mark the following numbers on thenumber line with a dot.

a 52 b 53 c 56 d 57

Draw a number line with 10 equal intervals marked from 0 to 10. Mark the numbers 2 and 7 on the number line with a dot.

THINK DRAW

Draw a straight line about 11 cm long. Put a small mark on the left-hand side of the line, leaving a -cm space from its end.Position a ruler along the line so that its 0 is at the first mark and put 10 more marks 1 cm apart. (Using a ruler will ensure that all intervals are equal.) Now write numbers from 0 to 10 underneath the marks.To mark a 2 and a 7, put dots directly above the marks with corresponding numbers on the number line.

1

12---

2

10 2 3 4 5 6 7 8 9 10

WORKEDExample

44

SkillSHEETanswersSkillSHEET 2.4Working with numbers on a number lineIf each mark on the number line represents a whole number (or an integer), then every number on the lineis 1 unit apart from its left-hand-side and right-hand-side ‘neighbours’.

Try theseReferring to the number line at right, which numbers are:

1 1 unit away from 6? . . . . . . . . . . . . and . . . . . . . . . . . .

2 1 unit away from 9? . . . . . . . . . . . . and . . . . . . . . . . . .

3 2 units away from 2? . . . . . . . . . . . . and . . . . . . . . . . . .

4 2 units away from 6? . . . . . . . . . . . . . and . . . . . . . . . . . .







Referring to the number line at right, which numbers are:

a 1 unit away from 2 b 3 units away from 6?

THINK WRITE

a Locate 2 on the number line. One unit away could be either one interval to the left or one interval to the right. Find such numbers on the line and write them down.

Numbers that are 1 unit away from 2 are 1 and 3.

b Locate 6 on the number line. Find the numbers that are 3 intervals to the left and to the right of 6 and write them down.

Numbers that are 3 units away from 6 are 3 and 9.

10 2 3 4 5 6 7 8 9 10

10 2 3 4 5 6 7 8 9 10

10 2 3 4 5 6 7 8 9 10

WORKEDExample

10 2 3 4 5 6 7 8 9 10

45

SkillSHEETanswersSkillSHEET 2.5Evaluating squares, cubes and cube rootsTo square a number means to write the number twice and multiply (or multiply the number by itself).

To cube a number means to write the number three times and multiply.

To find the cube root of a given number means to find a number that when written three times and multi-plied (that is, when cubed) will produce the given number. Note that cubing and finding the cube root aretwo inverse operations, which cancel each other out.

Try theseEvaluate each of the following.

1 32 = 3 × 3 2 62 = . . . . . . . . . . . . × 6 3 112 = . . . . . . . . . . . . × . . . . . . . . . . . .

32 = . . . . . . . . . . . . 62 = . . . . . . . . . . . . 112 = . . . . . . . . . . . .

4 92 = . . . . . . . . . . . . . . . . . . . . . . . . 5 72 = . . . . . . . . . . . . . . . . . . . . . . . . 6 52 = . . . . . . . . . . . . . . . . . . . . . . . .

32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . .

7 82 = . . . . . . . . . . . . . . . . . . . . . . . . 8 122 = . . . . . . . . . . . . . . . . . . . . . . . . 9 43 = 4 × 4 × 4

32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . .

10 33 = . . . . . . . . . . . . × 3 × . . . . . . . . . . . . 11 63 = . . . . . . . × . . . . . . . . × . . . . . . . 12 103 = . . . . . . . . . . . . . . . . . . . . . . . .

32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . .

13 53 = . . . . . . . . . . . . . . . . . . . . . . . . 14 73 = . . . . . . . . . . . . . . . . . . . . . . . . 15 13 = . . . . . . . . . . . . . . . . . . . . . . . .

32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . . 32 = . . . . . . . . . . . .

Evaluate each of the following.a 42 b 23 c

THINK WRITE

a To square a number means to multiply it by itself. So multiply 4 by itself to get the result.

a 42 = 4 × 442 = 16

b To cube a number means to write it three times and multiply. So multiply three lots of 2 together.

b 23 = 2 × 2 × 223 = 8

c To find a cube root of a given number means to find a number that when written three times and multiplied (cubed) will produce the given number. So find the number that when cubed equals 64. This number is your answer (since cube and cube root cancel each other out).

c =

= = 4

643

643 4 4× 4×3

433

WORKEDExample

46

SkillSHEETanswers16 203 = . . . . . . . . . . . . . . . . . . . . . . . . 17 = 18 =

203 = . . . . . . . . . . . . = =

= . . . . . . . . . . . . = . . . . . . . . . . . .

19 = 20 = 21 = . . . . . . . . . . . . . . . . . . . . . . . .

= = = . . . . . . . . . . . .

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

22 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 = . . . . . . . . . . . . . . . . . . . . . 24 = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

83 2 2× 2×3 1253... . . . . 5× . . . . . . .×3

.. . . . . .33 .. . . . . .

33

10003... . . . . . . . . . . .× . . . . . . .×3 273

.......................3 13

.. . . . . .33 .. . . . . .

33

2163 64 0003 3433

SkillSHEETanswers

47

SkillSHEETanswersSkillSHEET 2.6Simplifying algebraic terms written in an expanded formTo simplify a term written in an expanded form, multiply the numbers together and omit all multiplicationsigns. Note that it is conventional to place the resultant number (called a coefficient) in front of thepronumerals (letters).

Try theseSimplify each of the following terms.

1 2 × 3 × a 2 7 × 4 × c 3 3 × n × 4 4 a × 2 × 9 5 b × 4 × 4

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

6 6 × m × 7 7 5 × z × x 8 3 × x × 2 × y 9 4 × a × d × 8 10 5b × 2p

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

Simplify each of the following terms.a 5 × a × 3 b 2 × b × 7 × c

THINK WRITE

a Multiply 5 and 3 together, and place the resultant number (that is, 15) in front of the pronumeral a. Omit all multiplication signs.

a 5 × a × 3 = 15a

b Multiply 2 and 7 together. Write the resultant number (that is, 14) first, followed by the pronumerals b and c without any multiplication signs.

b 2 × b × 7 × c = 14bc

WORKEDExample

48

SkillSHEETanswersSkillSHEET 2.7Substitution into algebraic expressions ITo substitute the values of the pronumerals into an algebraic expression, replace all pronumerals with theirrespective values (or numbers). Once the values of the pronumerals are substituted, the expression can beevaluated. While evaluating, the correct order of operations must be observed.

Try theseEvaluate each algebraic expression if a = 2 and b = 3.

1 a + b 2 b − a 3 5ab

= . . . . . . . . . + . . . . . . . . . = . . . . . . . . . − . . . . . . . . . = . . . . . . . . . × . . . . . . . . . × . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

4 2ba 5 2a − b 6 5a + b

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

7 4a − 2b 8 3a2b 9 a + 4b2

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

10 ab ÷ 6

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

Evaluate each algebraic expression if a = 2 and b = 3.a a + 5b b 4ab

THINK WRITE

a Replace a with 2 and b with 3, and insert a multiplication sign between 5 and b (as 5b = 5 × b).

a a + 3b = 2 + 5 × 3

To evaluate, perform multiplication first, followed by addition.

a + 3b = 2 + 15a + 3b = 17

b Replace a with 2 and b with 3, and insert multiplication signs between the coefficient and the pronumerals.

b 4ab = 4 × 2 × 3

To evaluate, multiply all numbers together. 4ab = 24

1

2

1

2

WORKEDExample

49

SkillSHEETanswersSkillSHEET 2.8Substitution into algebraic expressions IIWhen evaluating algebraic fractions, substitute the values of the pronumerals first and then simplify theresultant fraction by dividing numerator and denominator by their highest common factor (HCF).Remember to change improper fractions into mixed numbers for your final answer.

Try theseEvaluate each of the following if a = 2, b = 3 and c = 6.

1 = 2 = 3 =

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

4 5 6

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

Evaluate each of the following if a = 2, b = 3 and c = 6.

a b

THINK WRITE

a Replace a with 2 and b with 3, and insert a multiplication sign between a and b (as ab = a × b).

a =

Evaluate the numerator of the fraction. =

Simplify the fraction by dividing the numerator and the denominator by their HCF of 2.

=

Convert an improper fraction into a mixed number.

= 1

b Replace a with 2, b with 3 and c with 6, and insert a multiplication sign between 2 and b (as 2b = 2 × b).

b =

Evaluate the numerator and the denominator of the fraction.

=

Simplify the fraction by dividing the numerator and the denominator by their HCF (2).

=

Convert the improper fraction into a mixed number.

= 1

ab4

------ a c+2b

------------

1ab4

------ 2 3×4

------------

264---

332---

4 12---

1a c+2b

------------ 2 6+2 3×------------

286---

343---

4 13---

WORKEDExample

ac--- 2

.......-------- b

c--- .......

6-------- ab

c------ 2 .......×

.......-----------------

acb

------ bca

------ abc------

SkillSHEETanswers

50

SkillSHEETanswers7 + b 8 + 2b 9

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

10

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

ac--- ac

12------ abc

3---------

ab c+5

---------------

51

SkillSHEETanswersSkillSHEET 2.9Order of operations IIIf an expression contains more than one operation, the calculations must be performed in the followingorder:1. brackets2. exponents or powers (for example, squares and cubes)3. multiplication and division (from left to right)4. addition and subtraction (from left to right).

Try theseCalculate each of the following, using the correct order of operations.

1 56 − 49 ÷ 7 2 3 × (14 − 7) 3 72 − 5 × 8

= 56 − . . . . . . . . . . . . = 3 × . . . . . . . . . . . . = . . . . . . . . . . . . − 5 × 8

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . − . . . . . . . . . . . .

= . . . . . . . . . . . .

Calculate each of the following, using the correct order of operations.a 7 + 8 ∏ 4 b 5 ¥ (2 + 4 ¥ 7) c 20 - 32 + 2

THINK WRITE

a Write the question.There are two operations to be performed: addition and division. Follow the order of operations rules: perform division first, followed by addition.

a 7 + 8 ÷ 4 = 7 + 2= 9

b Write the question.Calculations in brackets must be evaluated first. Inside the brackets there are two operations to be performed: addition and multiplication. According to the order of operations, multiplication must be done before the addition. Once the value of the expression in brackets is obtained, multiply it by 5.

b 5 × (2 + 4 × 7) = 5 × (2 + 28)= 5 × 30= 150

c Write the question. c 20 − 32 + 2 = 20 − 9 + 2Powers must be evaluated first. So start

with evaluating 32. (To square a number means to multiply it by itself, so 3 = 3 × 3 = 9.)Now there are two operations left: subtraction and addition. These are equally important and must be done in order of occurrence (that is, from left to right). So perform subtraction followed by addition.

= 11 + 2= 13

12

12

12

3

WORKEDExample

52

SkillSHEETanswers4 6 × 2 + 4 × 9 5 (6 + 4) × (10 − 3) 6 5 × 23 ÷ 10

= . . . . . . . . . . . . + . . . . . . . . . . . . × . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . + . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

7 4 + (40 − 8 × 4) 8 8 + 6 × 7 ÷ 2 9 (12 ÷ 3 − 4) × 100

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

10 19 + 8 ÷ 8 11 20 + 32 − 14 ÷ 2 12 22 + 6 × 8

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

53

SkillSHEETanswersSkillSHEET 2.10Operations with fractionsAdding and subtracting fractionsTo add or subtract fractions, follow these steps.1. Convert all mixed numbers to improper fractions.2. Convert (if necessary) to equivalent fractions with a common denominator.3. Add (subtract) numerators together and leave the denominator unchanged.4. Simplify (if necessary); if the result is an improper fraction, convert it to a mixed number.

Calculate each of the following.a + b - c 1 + 2

THINK WRITE

a The lowest common denominator (LCD) of 5 and 8 is 40. To obtain equivalent fractions with a denominator of 40, multiply the numerator and denominator of the first fraction by 8 and the second fraction by 5.

a + = × + ×

= +


=


= 1

b The LCD of 4 and 6 is 12. To obtain equivalent fractions with a denominator of 12, multiply the numerator and denominator of the first fraction by 3 and the second fraction by 2.

b − = × − ×

= −

Subtract the numerators (9 − 2) and leave the denominator (12) unchanged.

=

c Convert mixed numbers to improper fractions.

c 1 + 2 = +

The LCD of 2 and 8 is 8. To obtain an equivalent fraction with a denominator of 8, multiply the numerator and denominator of the first fraction by 4. (The denominator of the second fraction is already 8, so this fraction does not need to be changed.)

= × +

= +


=


= 4

25--- 7

8--- 3

4--- 1

6--- 1

2--- 5

8---

125--- 7

8--- 2

5--- 8

8--- 7

8--- 5

5---

1640------ 35

40------

25140------

31140------

134--- 1

6--- 3

4--- 3

3--- 1

6--- 2

2---

912------ 2

12------

2712------

112--- 5

8--- 3

2--- 21

8------

232--- 4

4--- 21

8------

128------ 21

8------

3338------

4 18---

WORKEDExample 1

54


a + b − c 1 + 1

= × + × = − × = +

= + = − = × + ×

= = = +

=

= . . . . . . . . . . . . . . . . . . . . . . . .

d + e − f 2 − 1

= × + × = × − × = −

= . . . . . . . . . . . .

+ . . . . . . . . . . . .

= . . . . . . . . . . . .

− . . . . . . . . . . . .

= × − ×

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . .

− . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

g + h − i + 2

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

.

15--- 3

8--- 5

8--- 1

2--- 2

3--- 1

8---

15--- .. .

8---- 3

8--- .. .

5---- 5

8--- 1

2--- 4

...---- 5

3--- .. .

8----

.. .

40------ .. .

40------ .. .

8---- .......

.......-------- 5

3--- .......

.......-------- 9

8--- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......--------

34--- 4

5--- 3

4--- 1

3--- 1

5--- 1

2---

34--- .......

.......-------- 4

5--- .......

.......-------- .......

.......-------- .......

.......-------- .. .

3---- .......

.......-------- 11

...------ .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

12--- 1

3--- 4

5--- 2

7--- 2

9--- 1

3---

55

SkillSHEETanswersj + k − l 3 − 1

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

Multiplying fractionsTo multiply fractions follow these steps.1. Convert all mixed numbers into improper fractions.2. Simplify as much as possible.3. Multiply numerators together and multiply denominators together.4. If the answer is an improper fraction, convert it to a mixed number.

710------ 7

8--- 5

6--- 1

8--- 1

4--- 1

3---

Perform the following multiplications.

a ¥ b 1 ¥

THINK WRITE

a Write the multiplication problem. a


=


b Write the multiplication problem. b 1

Convert the mixed number into an improper fraction. =

Cross-cancel 7 and 7 by dividing each by 7 (that is, 7 ÷ 7 = 1). Next cross-cancel 6 and 4 by dividing each by 2 (that is, 4 ÷ 2 = 2; 6 ÷ 2 = 3).

=

Multiply the numerators together and the denominators together.

=

Convert an improper fraction into a mixed number. = 1

35--- 1

6--- 3

4--- 6

7---

135--- 1

6---×

215--- 1

2---×

31

10------

134--- × 6

7---

274--- 6

7---×

312--- 3

1---×

432---

512---

WORKEDExample 2

56

SkillSHEETanswersTry these2 Perform each of the following multiplications.

a × b × c ×

= × = = ×

= =

d × e × f 1 ×

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

g × h × 2 i ×

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

j 1 × 2

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

25--- 1

2--- 1

3--- 1

2--- 3

10------ 1

3---

15--- .......

.......-------- .. .

6---- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

34--- 1

2--- 1

2--- 1

2--- 2

3--- 1

2---

78--- 10

3------ 4

5--- 1

4--- 4

9--- 18

4------

18--- 2

5---

57

SkillSHEETanswersDividing fractionsTo divide fractions follow these steps.1. Convert mixed numbers to improper fractions.2. Turn the second fraction upside down; replace ÷ with × and perform multiplication (tip and turn).3. If the result is an improper fraction, convert it to a mixed number.


a ÷ b ÷ c ÷

= × = × = ×

= × = = ×

= =


a ÷ b 1 ÷

THINK WRITE

a Turn the second fraction upside down and replace ÷ with ×. a =

Multiply numerators together and denominators together. =


b Convert the mixed number into an improper fraction first. b 1 =

Turn the second fraction upside down and replace ÷ with ×. =

Cross-cancel 4 and 2 by dividing each by 2 (i.e. 4 ÷ 2 = 2; 2 ÷ 2 = 1).

=



38--- 1

3--- 1

4--- 1

2---

138--- 1

3---÷ 3

8--- 3

1---×

298---

318---

114--- 1

2---÷ 5

4--- 1

2---÷

254--- 2

1---×

3 52--- 1

1---×

452---

512---

WORKEDExample 3

38--- 3

4--- 3

5--- 5

6--- 1

8--- 1

2---

38--- .. .

3---- 3

5--- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

58

SkillSHEETanswersd ÷ e ÷ f 1 ÷

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

g ÷ 1 h 1 ÷ i ÷ 1

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

j 2 ÷

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

49--- 1

3--- 5

9--- 2

3--- 1

4--- 3

8---

14--- 1

2--- 3

5--- 4

5--- 2

3--- 5

7---

34--- 1

6---

59

SkillSHEETanswersSkillSHEET 2.11Operations with decimalsAdding and subtracting decimalsWhen adding or subtracting decimals, add (or subtract) units to (from) units, tenths to (from) tenths, hun-dredths to (from) hundredths and so on. To set out addition (or subtraction), position the numbers so thatthe decimal points are in line (that is, directly under each other). If the numbers contain different numbersof digits after the decimal point, add zeros to make them equal in ‘length’ before adding (or subtracting).


a 5.6 + 12.9 b 13.9 + 4.76 c 190 + 4.58 d 0.855 + 2.6

15.6 13.90 190._ _ 0._ _ _

+ 121.9 + 4.76 + 4._ _ + 2._ _ _

_ _.5 _ _._ _ _ _ _._ _ _._ _ _

e 9.24 + 3.85 f 7 + 0.812 g 12.3 − 4.1 h 5.9 − 1.23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

+ . . . . . . . . . . . . . + . . . . . . . . . . . . − . . . . . . . . . . . . − . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Calculate each of the following.a 2.35 + 15.4 b 7.2 - 4.15

THINK WRITE

a Position the two numbers so that the decimal points are in line. Add a zero to the second number so that both numbers contain two digits after the decimal point. Add the numbers in columns (add 5 to 0, add 3 to 4 etc). The decimal point in the answer is directly below the decimal points in the question.

a 2.35+ 15.40

17.75

b Position the two numbers so that the decimal points are in line. Add one zero to the first number so that both numbers contain two digits after the decimal point. Start subtracting in columns. We cannot subtract 5 from 0, so we need to borrow from 2. Cross out 2 and write 1; 0 turns into 10. Now subtract (10 − 5 = 5, 1 − 1 = 0 and 7 − 4 = 3). The decimal point in the answer is directly below the decimal points in the question.

b 7.1210− 4.1115

3.1015

WORKEDExample 1

60

SkillSHEETanswersi 25.5 − 1.442 j 20 − 4.57 k 60.03 − 14.2 l 8.25 − 3.125

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

− . . . . . . . . . . . . − . . . . . . . . . . . . − . . . . . . . . . . . . − . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiplying decimalsTo multiply a decimal by a whole number, or by another decimal, first perform multiplication, ignoring thedecimal point(s) as you would if both were whole numbers. To find the position of the decimal point in theanswer, count the total number of decimal places in the numbers being multiplied. The number obtained forthe answer will have the same number of decimal places as the total number of decimal places in thenumbers being multiplied.

Note: When setting out multiplication, it is not necessary to position the numbers so that the decimalpoints are in line.

Try these2 Multiply each of the following.

a 7.9 × 6 b 6.5 × 5 c 1.79 × 7 d 6.2 × 0.5

57.9 6.5 1.79 . . . . . . . . . . . .

× 6 × 5 × 7 × . . . . . . . . . . . . .

__.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiply each of the following.a 9.78 ¥ 5 b 1.31 ¥ 3.2

THINK WRITE

a Multiply, ignoring the decimal point (that is, multiply 978 by 5). Count the numbers after the decimal point — there are two (7 and 8). Put the decimal point in the answer so that there are two numbers after it.

a 394.4748× 54484.4940

b Multiply, ignoring the decimal point (that is, multiply 131 by 32). Count all numbers after the decimal points — there are 3 (two in the first number and one in the second). Put the decimal point in the answer so that it has three decimal places.

b 1.3.1× 3.2

2.6.23.9.3.04.1.9.2

WORKEDExample 2

61

SkillSHEETanswerse 9.6 f 1.85 g 7.3 h 7.3

× 16 × 21 × 13 × 1.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i 8.9 j 9.42 k 64.5 l 5.72× 3.5 × 1.2 × 5.2 × 9.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dividing decimalsTo divide decimals, we need to change the second decimal (that is, the divisor) into a whole number. Thiscan be done by moving the decimal point to the right until it is at the end of the number. The first number(that is, the dividend) needs to be adjusted accordingly. That is, we need to move the decimal point in thedividend to the right as many places as we did in the divisor. When these adjustments are done, we canproceed with the division. Divide as you would for whole numbers and place the decimal point in theanswer directly above the decimal point in the question.

Calculate 4.598 ∏ 0.02.

THINK WRITE

To make the divisor (0.02) a whole number, move the decimal point 2 places to the right so that it is at the end of the number. Also move the decimal point in the dividend (4.598) 2 places to the right.

4.598 ÷ 0.02 = 459.8 ÷ 002.

Omit the zeros in front of the divisor and the decimal point at the end of it (that is, write 2 instead of 002.).

4.598 ÷ 0.23 = 459.8 ÷ 2

Set out the division. Divide as if both numbers are whole numbers. Place the decimal point in the answer directly above the decimal point in the question (that is, between 9 and 9).

2)212191.192)415191.18

1

2

3

WORKEDExample 3

62


a 9.236 ÷ 0.4 b 4.528 ÷ 0.2 c 52.38 ÷ 0.3 d 489.15 ÷ 0.5

= 92.36 ÷ 04. = 45.28 ÷ . . . . . . . . . . . . . = . . . . . . . . . . . . ÷ . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= 92.36 ÷ 4 = 45.28 ÷ . . . . . . . . . . . . . = . . . . . . . . . . . . ÷ . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

4)_ _ . _ _ . . . . . .)_ _ . _ _ . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

4)9 2 . 3 6 . . . . . .)4 5 . 2 8 . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

e 6.759 ÷ 0.02 f 5.2563 ÷ 0.03 g 24.928 ÷ 0.08 h 73.21 ÷ 0.04

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

i 4.581 ÷ 0.06 j 213.4 ÷ 0.04 k 171.6 ÷ 0.06 l 23.1 ÷ 0.11

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

m 10.056 ÷ 0.003 n 0.0004 ÷ 0.002 o 0.0255 ÷ 0.006 p 2.75 ÷ 0.005

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .). . . . . . . . . . . . . . . . . . . . . . . .

63

SkillSHEETanswersSkillSHEET 3.1Even and odd numbersEven numbers are numbers that end in 0, 2, 4, 6 or 8.Odd numbers are numbers that end in 1, 3, 5, 7 or 9.

Try theseFor each of the following, state whether the number is odd or even.

1 26 The number ends in 6, so the number is . . . . . . . . . . . . . .

2 73 The number ends in __, so the number is . . . . . . . . . . . . . .









For each of the following, state whether the number is odd or even.a 15 b 290

THINK WRITE

a The last digit of the number is 5, so state whether 15 is odd or even.

a 15 is an odd number.

b The number ends in 0, so state your conclusion.

b 290 is an even number.

WORKEDExample

64

SkillSHEETanswersSkillSHEET 3.2Consecutive numbersConsecutive numbers are numbers that follow one another. For example, 2, 4, 6 are consecutive evennumbers (since there are no other even numbers in between).

Try these

1 Write three consecutive odd numbers, beginning with 53. ____, ____, ____

2 Write four consecutive even numbers, beginning with 24. ____, ____, ____, ____

3 Write three consecutive prime numbers, beginning with 11. ____, ____, ____

4 Write five consecutive square numbers, beginning with 9. ____, ____, ____, ____, ____

5 Write three consecutive multiples of 5, beginning with 25. ____, ____, ____

6 Write three consecutive multiples of 7, of which 14 is the smallest. ____, ____, ____

7 Write three consecutive odd numbers, of which 121 is the smallest. ____, ____, ____

8 Write five consecutive multiples of 8, of which 56 is the largest. ____, ____, ____, ____, ____

9 Write four consecutive even multiples of 3, of which 12 is the smallest. ____, ____, ____, ____

10 Write three consecutive odd multiples of 5, of which 105 is the largest. ____, ____, ____

a Write three consecutive odd numbers, beginning with 11.b Write three consecutive multiples of 6, of which 12 is the smallest.

THINK WRITE

a Start with 11. Write the next odd number after 11 and the one that follows.

a 11, 13, 15

b Since 12 is the smallest, begin with 12. Write the next two multiples of 6 (this can be done by simply adding 6 to 12 and then to 18).

b 12, 18, 24

WORKEDExample

65

SkillSHEETanswersSkillSHEET 3.3Square numbersTo find the square of a number, multiply the number by itself.Note: Square numbers can be written using an index of 2, for example 42.


1 Square of 3 2 Square of 6 3 Square of 8

= 3 × 3 = 6 × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × 8

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . . = . . . . . . . . . . . . . .


= . . . . . . . . . . . . . . × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . . = . . . . . . . . . . . . . .


= . . . . . . . . . . . . . . × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

10 Square of 7

= . . . . . . . . . . . . . . × . . . . . . . . . . . . . .

= . . . . . . . . . . . . . .

Find the square of 4, that is, 4 ¥ 4.

THINK WRITE

To square a number, multiply it by itself, so multiply 4 by 4.

Square of 4 = 4 × 4Square of 4 = 16

WORKEDExample

66

SkillSHEETanswersSkillSHEET 3.4MultiplesA multiple is the result of multiplication of one number by another number. For example, 10 is a multipleof both 2 and 5, since 2 × 5 = 10.

Try theseWrite the first five multiples of each of the following numbers.











Write the first five multiples of 2.

THINK WRITE

To find the first five multiples of 2, multiply 2 by 1, then by 2, 3, 4 and finally by 5. Each product (that is, the result of each multiplication) is a multiple of 2.

2 × 1 = 22 × 2 = 42 × 3 = 62 × 4 = 82 × 5 = 10

List the first five multiples of 2. The first five multiples of 2 are: 2, 4, 6, 8, 10.

1

2

WORKEDExample

67

SkillSHEETanswersSkillSHEET 3.5Prime and composite numbersA prime number is a number which has only two factors — itself and 1. A composite number is a numberthat has other factors besides itself and 1.

The number 1 is neither prime nor composite. Its only factor is itself (1 = 1 × 1).

Try theseFor each of the following, list the factors and state whether the number is prime or composite.

1 The factors of the number 5 are . . . . . . . . . . . . . . . . . . . . . . . . , so the number 5 is . . . . . . . . . . . . . . . . . . . . . . . . .










State whether the following numbers are prime or composite.a 13 b 12

THINK WRITE

a Find all the factors of 13. a 13 = 13 × 1There are no other factors, so 13 is a prime number.

b Find all the factors of 12 b 12 = 1 × 1212 = 2 × 612 = 3 × 4The factors of 12 are 1, 2, 3, 4, 6 and 12. This means that 12 is a composite number.

WORKEDExample

SkillSHEETanswers

68

SkillSHEETanswersSkillSHEET 3.6Factor pairsA number that divides exactly into another number is a factor of that number. The result of the division(that is, the quotient) is also a factor. These two factors form a factor pair. For example, if 7 is a factor of35, then 5 is also a factor of 35 because 35 ÷ 7 = 5, and so 5 and 7 form a pair of factors of 35.

The product of the numbers in a factor pair is always equal to the number whose factors they are. Forexample, if 3 and 6 is a factor pair of 18, then 3 × 6 = 18.

Note: There is an alternative way of finding a missing factor. For example, if 2 is a factor of 12, we canobtain the other factor by finding the number that will give 12 when multiplied by 2. That is, since 2 × 6 = 12,2 and 6 form a pair of factors of 12.

Try these1 Find the missing factor in each of the following pairs of factors of 24:

3 and ____ , 6 and ____ , 12 and ____ .

2 Find the missing factor in each of the following pairs of factors of 39:

3 and ____ , 39 and ____ .





Find the missing factor in each of the following factor pairs of 12:2 and __ , 4 and __ .

THINK WRITE

To find the missing factor, divide 12 by the known factor of the pair.

12 ÷ 2 = 6So a factor pair of 12 is 2 and 6.

Repeat step 1 for the second pair. 12 ÷ 4 = 3So a factor pair of 12 is 4 and 3.

1

2

WORKEDExample

SkillSHEETanswers

69

SkillSHEETanswersSkillSHEET 3.7Perfect cubesTo find the cube of a number, we write the number three times and multiply.Note: Cube numbers can be written using an index of 3; for example, 43.


1 The cube of 3 2 The cube of 5

= 3 × 3 × 3 = 5 × 5 × . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .


= . . . . . . . . . . . . . . × 10 × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × . . . . . . . . . . . . . . × . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .


= . . . . . . . . . . . . . . × . . . . . . . . . . . . . . × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × . . . . . . . . . . . . . . × . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .


= . . . . . . . . . . . . . . × . . . . . . . . . . . . . . × . . . . . . . . . . . . . . = . . . . . . . . . . . . . . × . . . . . . . . . . . . . . × . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

Find the cube of 4; that is, 4 ¥ 4 ¥ 4.

THINK WRITE

To cube a number we need to write the number three times and then multiply.

The cube of 4 = 4 × 4 × 4The cube of 4 = 64

WORKEDExample

70

SkillSHEETanswersSkillSHEET 3.8Disjoint and overlapping setsDisjoint sets are those that have no elements in common. Overlapping sets have some elements in common.

Try theseFor each of the following, state whether the sets are disjoint or overlapping.

1 ξ = {standard deck of playing cards}, A = {queens}, B = {number cards}

These sets are . . . . . . . . . . . . . . . . . . . . . . . . because . . . . . . . . . . . . . . . . . . . . . . . . .

2 ξ = {standard deck of playing cards}, A = {hearts}, B = {cards numbered 10}


3 ξ = {standard deck of playing cards}, A = {numbered cards}, B = {picture cards}


4 ξ = {standard deck of playing cards}, A = {aces}, B = {diamonds}


5 ξ = {numbers on a standard 6-sided die}, A = {odd numbers}, B = {multiples of 3}


6 ξ = {numbers on a standard 6-sided die}, A = {5}, B = {factors of 6}


7 ξ = {numbers on a standard 6-sided die}, A = {even numbers}, B = {square numbers}


8 ξ = {numbers on a standard 6-sided die}, A = {even numbers}, B = {multiples of 3}


9 ξ = {numbers on a standard 6-sided die}, A = {even numbers}, B = {prime numbers}


10 ξ = {faces on a coin}, A = {Tails}, B = {Heads}


ξ = {standard deck of playing cards} and B = {kings}State whether the following sets are disjoint or overlapping.a A = {diamonds} and B = {kings}b A = {clubs} and B = {spades}

THINK WRITE

a Are there any elements common to both set A and set B?

a There is a king of diamonds, so there is an element common to both sets. This means that the sets are overlapping.

b Are there any elements common to both set A and set B?

b A card cannot be both a club and a spade, so there are no common elements: the sets are disjoint.

WORKEDExample

71

SkillSHEETanswersSkillSHEET 3.9Venn diagramsVenn diagrams comprise a rectangle to represent the sample space and a series of circles to represent sets ofelements.

The area common to two or more sets is referred to as the intersection of these sets.Note: Either ε or ξ is used to represent the universal set.

Try theseInterpret each of the following Venn diagrams by stating what each shaded area represents

1 2 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Interpret each of the following Venn diagrams by stating what each shaded area represents.a b

THINK WRITE

a Study the diagram. The shaded area is common to both sets, so it is their intersection.

a Shaded area represents the intersection of sets A and B.

b Study the diagram. Set B is shaded, but not entirely. Only the part of set B that is not shared with sets A and C is shaded.

b Shaded area represents set B only. (That is, set B but not sets A and C.)

A B

εA B

C

ε

WORKEDExample

A B

εA B

εA B

ε

A B

εA B

εA B

ε

SkillSHEETanswers

72

SkillSHEETanswers7 8 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 11 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B C

Aε

A B

C

εA B

C

ε

A B

C

εA B

C

ε

B C

Aε

73

SkillSHEETanswersSkillSHEET 3.10Set notationA set is a collection of similar elements.

The universal set (or sample space) is the largest set, containing all possible elements. It is denoted by theletter ξ.

The intersection of sets is represented by the elements present in each set (common elements). It isdenoted by the sign ∩, and is usually associated with the word and.

The union of the sets is represented by the combined set containing all elements present in every set (withcommon elements written once). It is denoted by the sign ∪, and is usually associated with the word or.(Note: When we say ‘A or B’ we mean ‘A or B, or both’.)

The complement of a set contains all elements that are outside the set (that is, the elements that are in ξ,but not in a given set itself). If the original set is denoted by A, its complement is denoted as A′.

The number of elements in any set is denoted by the letter n. For example, n (A) means ‘the number ofelements in set A’.

Try theseIf ξ = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, A = {10, 12, 14, 16, 18, 20}, B = {11, 13, 15, 17, 19} andC = {12, 15, 18}, find:

1 n(ξ) 2 n(A) 3 n(C) 4 n (A ∪ C) 5 n(B ∩ C)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 A′ 7 C ′ 8 A ∩ C 9 B ∪ C 10 C ′ ∩ A

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

If ξ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {0, 2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9} and C = {3, 6, 9}, find:a n(A) b n(B « C) b A » C b B¢

THINK WRITE

a n(A) is the number of elements in set A. So count the number of elements in A and write the answer.

a n(A) = 6

b n(B ∩ C) means the number of elements common to both B and C. Observe that the elements common to B and C are 3 and 9. Write the answer.

b n(B ∩ C) = 2

c A ∪ C means the set of elements that are in set A, set C and in both sets. Write down all such elements. The elements that appear in both sets (6) should be written only once.

c A ∪ C = {0, 2, 3, 4, 6, 8, 9, 10}

d B′ means all elements that are in the universal set but not in set B. List all such elements. (Notice that B′ = A)

d B′ = {0, 2, 4, 6, 8, 10}

WORKEDExample

74

SkillSHEETanswersSkillSHEET 4.1Simplifying fractions with a denominator of 100To simplify a fraction, divide both the numerator and the denominator by their highest common factor.Remember that the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100.

Try theseSimplify each of the following.

1 = 2 = 3 = . . . . . . . . . . . .

4 = . . . . . . . . . . . .

5 = . . . . . . . . . . . .

6 = . . . . . . . . . . . .

= . . . . . . . . . . . .

7 = . . . . . . . . . . . .

8 = . . . . . . . . . . . .

9 = . . . . . . . . . . . .

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

10 = . . . . . . . . . . . .

= . . . . . . . . . . . .

Simplify each of the following.

a b

THINK WRITE

a The highest common factor of 8 and 100 is 4, so divide the numerator and the denominator by 4.

a =

b First convert the improper fraction into a mixed numeral.

b = 2

Simplify the fractional part by dividing both the numerator and the denominator by 2.

= 2

8100--------- 226

100---------

8100--------- 2

25------

1226100--------- 26

100---------

21350------

WORKEDExample

80100--------- 4

...---- 6

100--------- .......

.......-------- 24

100---------

45100--------- 75

100--------- 260

100---------

502100--------- 416

100--------- 125

100---------

315100---------

SkillSHEETanswers

75

SkillSHEETanswersSkillSHEET 4.2Converting a mixed number into an improper fractionTo convert a mixed number into an improper fraction, follow these steps.1. Multiply the denominator by the whole part and add the numerator. 2. Put the resultant number as the numerator of the improper fraction. 3. Write the denominator. (It is the same as the one in the mixed number.)


1 1 = 2 2 = 3 1 = . . . . . . . . . . . .

4 2 = . . . . . . . . . . . .

5 3 = . . . . . . . . . . . .

6 2 = . . . . . . . . . . . .

7 4 = . . . . . . . . . . . .

8 1 = . . . . . . . . . . . .

9 5 = . . . . . . . . . . . .

10 4 = . . . . . . . . . . . .


THINK WRITE


2 =

38---

38--- 19

8------

WORKEDExample

35---

. . .

5---- 5

8---

21

...------ 2

7---

78--- 1

2--- 4

5---

34--- 1

9--- 5

6---

37---

SkillSHEETanswers

76

SkillSHEETanswersSkillSHEET 4.3Multiplying decimals by powers of 10To multiply a decimal by a power of 10, move the decimal point to the right one space for each zero in thepower of 10. For example, to multiply by 10 move the decimal point one place to the right, and to multiplyby 1000 move it three places to the right. Note that if there are not enough digits after the decimal point, wecan always add extra zeros.


1 2.56 × 10 = . . . . . . . . . . . . 2 7.6 × 10 = . . . . . . . . . . . .

3 0.98 × 10 = . . . . . . . . . . . . 4 3.49 × 100 = . . . . . . . . . . . .

5 2.6 × 100 = . . . . . . . . . . . . 6 70.1 × 100 = . . . . . . . . . . . .

7 0.2 × 100 = . . . . . . . . . . . . 8 5.321 × 1000 = . . . . . . . . . . . .

9 10.2 × 1000 = . . . . . . . . . . . . 10 0.758 × 1000 = . . . . . . . . . . . .

11 2.5 × 10 000 = . . . . . . . . . . . . 12 3.576 × 10 000 = . . . . . . . . . . . .

13 0.003 × 1000 = . . . . . . . . . . . . 14 0.000 6 × 10 000 = . . . . . . . . . . . .

15 0.000 08 × 100 000 = . . . . . . . . . . . . 16 0.04 × 10 000 = . . . . . . . . . . . .

Calculate each of the following.a 5.67 × 10 b 0.7 × 100

THINK WRITE

a To multiply a decimal by 10, move the decimal point one place to the right (as there is one zero in 10).

a 5.67 × 10 = 56.7

b To multiply a decimal by 100, we need to move the decimal point two places to the right. However, there is only one digit after the decimal (7). So add a zero first (to create two decimal places), and then move the decimal point two places to the right. Note that we write the answer as 70, rather than 070.

b 0.7 × 100

= 0.70 × 100= 70

WORKEDExample

77

SkillSHEETanswersSkillSHEET 4.4Dividing whole numbers and decimals by 100To divide a whole or a decimal number by 100, move the decimal point two places to the left. Note thatalthough a whole number does not have a decimal point, we can always add it at the end of the number.(For example, 35 and 35. are the same numbers.) Also note that if there are not enough digits to move thedecimal point the required number of places, we can always add extra zeros.


a 28 ÷ 100 = . . . . . . . . . . . . b 60 ÷ 100 = . . . . . . . . . . . . c 34 ÷ 100 = . . . . . . . . . . . .

d 2 ÷ 100 = . . . . . . . . . . . . e 15 ÷ 100 = . . . . . . . . . . . . f 7 ÷ 100 = . . . . . . . . . . . .

g 560 ÷ 100 = . . . . . . . . . . . . h 721 ÷ 100 = . . . . . . . . . . . . i 3 ÷ 100 = . . . . . . . . . . . .

j 75 ÷ 100 = . . . . . . . . . . . . k 600 ÷ 100 = . . . . . . . . . . . . l 250 ÷ 100 = . . . . . . . . . . . .

2 Calculate each of the following.

a 9.2 ÷ 100 = . . . . . . . . . . . . b 52.3 ÷ 100 = . . . . . . . . . . . . c 0.5 ÷ 100 = . . . . . . . . . . . .

d 8.19 ÷ 100 = . . . . . . . . . . . . e 4.9 ÷ 100 = . . . . . . . . . . . . f 123.4 ÷ 100 = . . . . . . . . . . . .

g 0.3 ÷ 100 = . . . . . . . . . . . . h 71.1 ÷ 100 = . . . . . . . . . . . . i 155.6 ÷ 100 = . . . . . . . . . . . .

j 4.25 ÷ 100 = . . . . . . . . . . . . k 75.3 ÷ 100 = . . . . . . . . . . . . l 100.5 ÷ 100 = . . . . . . . . . . . .

Calculate each of the following.a 34 ÷ 100 b 350 ÷ 100 c 75.6 ÷ 100 d 4.1 ÷ 100

THINK WRITE

a Put a decimal point at the end of the whole number.

a 34 ÷ 100 = 34. ÷ 100

To divide by 100, move the decimal point 2 places to the left. Add a zero in front of the decimal point.

34 ÷ 100 = 0.34

b Put a decimal point at the end of the whole number.

b 350 ÷ 100 = 350. ÷ 100

Move the decimal point 2 places to the left. You may wish to rewrite your answer, omitting the zero at the end of the resulting decimal (as 3.50 = 3.5).

350 ÷ 100 = 3.50350 ÷ 100 = 3.5

c Move the decimal point 2 places to the left and add a zero in front of it.

c 75.6 ÷ 100 = 0.756

d Move the decimal point two places to the left. (Add some extra zeros in front of the number as you go.)

d 4.1 ÷ 100 = 0.041

1

2

1

2

WORKEDExample

78

SkillSHEETanswersSkillSHEET 4.5Changing fractions to equivalent fractions with a denominator of 100To change a fraction into an equivalent fraction with a denominator of 100:1. establish how many times the given denominator fits (divides) into 1002. multiply both the numerator and the denominator by that number.

Remember that factor pairs of 100 could be of assistance. These are 1 and 100, 2 and 50, 4 and 25, 5 and20, and 10 and 10. So if the given denominator is one factor in a pair, then both the numerator and thedenominator must be multiplied by the other factor in that pair. For example, if the given denominator is 20,the numerator and the denominator must be multiplied by 5.

Try theseChange each of the following into equivalent fractions with a denominator of 100.

1 = 2 = 3 =

= = =

4 = 5 = 6 =

= = =

7 = 8 = 9 =

= = =

Change into an equivalent fraction with a denominator of 100.

THINK WRITE

The denominator of the given fraction (that is, 5) fits into 100 twenty times. So multiply both numerator and denominator by 20.

=

=

35---

35--- 3 20×

5 20×---------------

60100---------

WORKEDExample

15---

1 .......×5 .......×----------------- 4

5---

....... .......×

....... .......×----------------------- 3

10------

....... .......×

....... .......×-----------------------

.......

100--------- .......

.......-------- .......

100---------

710------

....... .......×

....... .......×----------------------- 1

2---

....... .......×

....... .......×----------------------- 3

4---

....... .......×

....... .......×-----------------------

.......

.......-------- .......

.......-------- .......

.......--------

320------

....... .......×

....... .......×----------------------- 8

25------

....... .......×

....... .......×----------------------- 39

50------

....... .......×

....... .......×-----------------------

.......

.......-------- .......

.......-------- .......

.......--------

79

SkillSHEETanswersSkillSHEET 4.6Multiplying fractions by 100To multiply a fraction by 100:1. write 100 as a fraction, by placing it over 12. cancel where possible3. multiply numerators together and denominators together4. convert to a mixed number if necessary.

Try theseMultiply each of the following fractions by 100.

1 2 3

× × ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

4 5 6

× × ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

Multiply by 100.

THINK WRITE

Set out the multiplication by placing 100 over 1. ×

Simplify by dividing both 15 and 100 by 5. = ×


=

Convert to a mixed number. = 26

415------

14

15------ 100

1---------

243--- 20

1------

3803

------

423---

WORKEDExample

20

3

410------ 8

20------ 12

50------

410------ 100

1--------- 8

20------ 100

1--------- 12

50------ .......

.......--------

41--- .......

1-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

35--- 14

25------ 8

15------

35--- .......

.......-------- 14

25------ .......

.......-------- 8

15------ .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

80


× × ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

10

×

= ×

=

= . . . . . . . . . . . .

1740------ 14

30------ 27

45------

1740------ .......

.......-------- 14

30------ .......

.......-------- 27

45------ .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

1570------

1570------ .......

.......--------

.......

.......-------- .......

.......--------

.......

.......--------

81

SkillSHEETanswersSkillSHEET 4.7Converting an improper fraction into a mixed numberTo convert an improper fraction into a mixed number, divide the numerator by the denominator. The quo-tient will be the whole part of the mixed number and the remainder will be the numerator of the fractionalpart. (The denominator of the fractional part is the same as the denominator of the improper fraction.)

Try theseChange each of the following improper fractions into a mixed number.

1 = 1 2 = 2 3 = . . . . . . . . . . . .

4 = . . . . . . . . . . . .

5 = . . . . . . . . . . . .

6 = . . . . . . . . . . . .

7 = . . . . . . . . . . . .

8 = . . . . . . . . . . . .

9 = . . . . . . . . . . . .

10 = . . . . . . . . . . . .

Change into a mixed number.

THINK WRITE

Divide 7 by 5. Five goes into seven once with a remainder of 2. So the whole part of the mixed number is 1, and the numerator of the fractional part is 2. (The denominator remains unchanged.)

= 1

75---

75--- 2

5---

WORKEDExample

95--- .......

5-------- 7

3--- .......

.......-------- 8

3---

175

------ 127

------ 143

------

174

------ 236

------ 425

------

358

------

82

SkillSHEETanswersSkillSHEET 4.8Multiplying decimals by 100To multiply a decimal by 100, move the decimal point two places to the right. Note that if there are notenough digits after the decimal point, we can always add extra zeros.


1 0.56 × 100 = . . . . . . . . . . . . 2 0.76 × 100 = . . . . . . . . . . . . 3 0.98 × 100 = . . . . . . . . . . . .

4 0.49 × 100 = . . . . . . . . . . . . 5 2.6 × 100 = 2.60 × 100 6 1.1 × 100 = . . . . . .

× 100

= . . . . . . . . . . . . = . . . . . . . . . . . .

7 0.2 × 100 = . . . . . . . . . . . . 8 5.321 × 100 = . . . . . . . . . . . . 9 10.2 × 100 = . . . . . . . . . . . .

= . . . . . . . . . . . . = . . . . . . . . . . . .

10 0.758 × 100 = . . . . . . . . . . . . 11 0.5 × 100 = . . . . . . . . . . . . 12 0.0006 × 100 = . . . . . . . . . . . .

= . . . . . . . . . . . .


THINK WRITE

a To multiply a decimal by 100, move the decimal point two places to the right. Note that we do not write the zero in front of the number and the decimal point at the end of the number (i.e. we write 67 rather than 067).

a 0.67 × 100 = 67

b We need to move the decimal point two places to the right; however, there is only one digit after the decimal point (7). So add a zero first (to create two decimal places), and then move the decimal point.

b 0.7 × 100

= 0.70 × 100= 70

WORKEDExample

83

SkillSHEETanswersSkillSHEET 4.9Multiplying fractions by a whole numberTo multiply a fraction by a whole number:1. change the whole number into a fraction by writing it over 12. simplify as much as possible3. multiply the numerators together and the denominators together4. if the answer is an improper fraction, convert it to a mixed number.

Perform each of the following multiplications.a × 360 b × 25

THINK WRITE

a Write the given problem. a × 360

Convert 360 into a fraction by writing it over 1.

= ×

Cancel down 60 and 100 by dividing each by 20 (that is, 60 ÷ 20 = 3; 100 ÷ 20 = 5).

= ×


= ×


=

Convert the improper fraction into a mixed number (which in this case is actually a whole number).

= 216

b Write the given problem. b × 25


= ×


= ×


= ×


=


= 11

60100--------- 45

100---------

160100---------

260100--------- 360

1---------

335--- 360

1---------

4 31--- 72

1------

52161

---------

6

145100---------

245100--------- 25

1------

3920------ 25

1------

4 94--- 5

1---

5454------

6 14---

WORKEDExample

84

SkillSHEETanswersTry thesePerform each of the following multiplications.

1 × 36 2 × 58 3 × 20

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

4 × 32 5 × 70 6 × 160

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

7 × 5000 8 × 250 9 × 80

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

10 × 75 11 × 24 12 × 40

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

20100--------- 10

100--------- 35

100---------

.......

100--------- 36

.......-------- 10

.......-------- .......

1-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

75100--------- 5

100--------- 2

100---------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

17100--------- 92

100--------- 45

100---------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

110100--------- 230

100--------- 125

100---------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

85

SkillSHEETanswersSkillSHEET 4.10Multiplying decimals by whole numbers and decimalsTo multiply a decimal by a whole number, or by another decimal, first perform the multiplication, ignoringthe decimal point(s), as you would if both numbers were whole. To find the position of the decimal point,count the total number of decimal places in the question and then put the decimal point in the answer sothat it (the answer) has the same number of decimal places.

Note that when setting out multiplication, it is not necessary to position the numbers so that the decimalpoints are in line.

Try theseMultiply each of the following.

1 0.79 × 6 2 0.65 × 5 3 1.79 × 7 4 0.73 × 13

0.79 0.65 1.79 0.73× 6 × 5 × 7 × 13

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

5 0.96 × 16 6 1.85 × 21 7 0.62 × 0.5 8 0.03 × 1.7

0.96 1.85 0.62 0.03× 16 × 21 × 0.5 × 1.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiply each of the following.a 0.78 × 5 b 1.31 × 3.2

THINK WRITE

a Multiply, ignoring the decimal point (that is, multiply 78 by 5). Count the digits after the decimal point; there are two (7 and 8). Put the decimal point in the answer so that there are two digits after it.

a × 30.478× 03.375× 3.390

b Multiply, ignoring the decimal point (that is, multiply 131 by 32). Count all digits after the decimal points; there are 3 (two in the first number and one in the second). Put the decimal point in the answer so that it creates three decimal places.

b × 1.31× 3.2× . 262× .3930× 4.192

WORKEDExample

86

SkillSHEETanswers9 0.9 × 3.5 10 0.4 × 1.2 11 0.55 × 5.2 12 0.04 × 9.1

0.9 0.4 0.55 0.04× 3.5 × 1.2 × 5.2 × 9.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

SkillSHEETanswersSkillSHEET 4.11Rounding to the nearest whole numberTo round to the nearest whole number, consider the digit in the tenths’ place (that is, the first digit after thedecimal point).1. If it is smaller than 5, write the whole part of the number as it is and omit the decimal part.2. If it is 5 or larger, increase the whole part by 1 and omit the decimal part.

Try theseRound each of the following to the nearest whole number.

1 23.41 ≈ . . . . . . . . . . . . . . 2 7.82 ≈ . . . . . . . . . . . . . . 3 12.07 ≈ . . . . . . . . . . . . . . 4 10.88 ≈ . . . . . . . . . . . . . .

5 56.24 ≈ . . . . . . . . . . . . . . 6 122.39 ≈ . . . . . . . . . . . . . . 7 78.521 ≈ . . . . . . . . . . . . . . 8 99.423 ≈ . . . . . . . . . . . . . .

9 9.62 ≈ . . . . . . . . . . . . . . 10 18.183 ≈ . . . . . . . . . . . . . .

Round each of the following to the nearest whole number.a 2.35 b 15.81

THINK WRITE

a The first digit after the decimal point is 3. Since 3 is smaller than 5, leave the whole part of the number unchanged (that is, a 2) and omit the decimal part.

a 2.35 ≈ 2

b Here the digit in the tenths’ place is 8, which is greater than 5. Add 1 to the whole part of the number (15 + 1 = 16), and leave out the decimal part.

b 15.81 ≈ 16

WORKEDExample

88

SkillSHEETanswersSkillSHEET 4.12Rounding decimals to 2 decimal placesTo round a decimal correct to 2 decimal places, follow these steps.1. Consider the digit in the third decimal place (that is, the thousandths’ place).2. If it is less than 5, simply omit this digit and all digits that follow. (That is, omit all digits beginning from

the third decimal place.) 3. If it is 5 or greater than 5, add 1 to the preceding digit (that is, the one in the hundredth’s place) and omit

all digits beginning from the third decimal place.

Note that the sign ≈ is read as ‘is approximately equal to’.

Try theseRound each of the following numbers correct to 2 decimal places.

1 0.322 ≈ . . . . . . . . . . . . . . 2 0.257 ≈ . . . . . . . . . . . . . . 3 1.723 ≈ . . . . . . . . . . . . . . 4 2.555 ≈ . . . . . . . . . . . . . .

5 4.308 ≈ . . . . . . . . . . . . . . 6 12.195 ≈ . . . . . . . . . . . . . . 7 8.4678 ≈ . . . . . . . . . . . . . . 8 25.033 78 ≈ . . . . . . . . . . . .

9 18.333 333 ≈ . . . . . . . . . . . . . . 10 0.166 666 6 ≈ . . . . . . . . . . . . . .

Round each of the following numbers correct to 2 decimal places.a 0.239 b 4.5842

THINK WRITE

a The digit in the third decimal place is 9, which is greater than 5. So add 1 to the preceding digit (that is, to 3) and omit 9.

a 0.239 ≈ 0.24

b The digit in the third decimal place is 4. Since it is less than 5, simply omit all digits beginning from the third decimal place (that is, omit 4 and 2).

b 4.5842 ≈ 4.58

WORKEDExample

89

SkillSHEETanswersSkillSHEET 4.13Rounding money to the nearest 5 centsTo round money to the nearest 5 cents, follow these rules.1. Round down the amounts ending in 1c and 2c and round up the amounts ending in 8c and 9c to the

nearest 10c.2. Round up the amounts ending in 3c and 4c and round down those ending in 6c and 7c to the nearest 5c.

Try these

Round each of the following amounts to the nearest 5 cents.

1 21c ≈ . . . . . . . . . . . . . . 2 87c ≈ . . . . . . . . . . . . . . 3 56c ≈ . . . . . . . . . . . . . . 4 72c ≈ . . . . . . . . . . . . . .

5 48c ≈ . . . . . . . . . . . . . . 6 33c ≈ . . . . . . . . . . . . . . 7 64c ≈ . . . . . . . . . . . . . . 8 69c ≈ . . . . . . . . . . . . . .

9 27c ≈ . . . . . . . . . . . . . . 10 91c ≈ . . . . . . . . . . . . . .

Round each of the following amounts to the nearest 5 cents.a 31c b 56c c 78c d 83c

THINK WRITE

a The amount ends with 1, so round it down to the nearest 10c.

a 31c ≈ 30c

b The amount ends with 6, so round it down to the nearest 5c

b 56c ≈ 55c

c The amount ends with 8, so round it up to the nearest 10c.

c 78c ≈ 80c

d The amount ends with 3, so round it up to the nearest 5c.

d 83c ≈ 85c

WORKEDExample

90

SkillSHEETanswersSkillSHEET 5.1Alternative expressions used to describe the four operationsThe four operations can be described using different words or expressions. The table below shows the mostcommon ways of specifying the operation needed.

Try these

Write each of the following as a mathematical sentence.

1 The sum of 3 and 5 . . . . . . . . . . . . . . . . . . . . . . . . 2 5 less than 8 . . . . . . . . . . . . . . . . . . . . . . . .

3 4 is added to 7 . . . . . . . . . . . . . . . . . . . . . . . . 4 3 taken away from 12 .. . . . . . . . . . . . . . . . . . . . . . .

5 5 increased by 2 . . . . . . . . . . . . . . . . . . . . . . . . 6 4 is multiplied by 5 . . . . . . . . . . . . . . . . . . . . . . . .

7 4 subtracted from 9 . . . . . . . . . . . . . . . . . . . . . . . . 8 The difference between 5 and 1 . . . . . . . . . . . . . . . . . . . . . . . .

9 The product of 2 and 7 . . . . . . . . . . . . . . . . . . . . . . . . 10 The quotient of 9 and 3 . . . . . . . . . . . . . . . . . . . . . . . .

11 9 more than 8 . . . . . . . . . . . . . . . . . . . . . . . . 12 12 decreased by 5 . . . . . . . . . . . . . . . . . . . . . . . .

13 7 times 6 . . . . . . . . . . . . . . . . . . . . . . . . 14 10 divided by 5 . . . . . . . . . . . . . . . . . . . . . . . .

Operation symbol

Words/expressions used to describe the operation Example

Mathematical sentence

+

Plus Add (or ‘added to’)The sum ofIncrease (or ‘increased by’)More than

5 plus 35 is added to 3The sum of 5 and 3Increase 5 by 33 more than 5

5 + 3

−

MinusSubtract (or ‘subtracted from’)Take away (or ‘taken away from’)The difference betweenLess thanDecrease (or ‘decreased by’)

7 minus 2Subtract 2 from 72 is taken away from 7The difference between 7 and 22 less than 7Decrease 7 by 2

7 − 2

× Multiply (or ‘multiplied by’)The product of… times

Multiply 3 by 4The product of 3 and 44 times 3

3 × 4

÷ Divide (or ‘divided by’)ShareQuotient

Divide 8 by 2Share 8 between 2The quotient of 8 and 2

8 ÷ 2

Write each of the following as a mathematical sentence.a The sum of 4 and 5 b 12 is divided by 4

THINK WRITE

a The expression ‘the sum of’ is represented by +, so write a mathematical sentence by joining given numbers with the + sign.

a 4 + 5

b The expression ‘divided by’ is represented by ÷. The number which is to be divided is written first, and the number by which we need to divide is written second.

b 12 ÷ 4

WORKEDExample

91

SkillSHEETanswersSkillSHEET 5.2Order of operations IIIf an expression contains more than one operation, the calculations must be performed in the followingorder:1. brackets2. multiplication and division (from left to right)3. addition and subtraction (from left to right).

Try theseFind the value of each of the following, using the order of operations rules.1 56 − 49 ÷ 7 2 18 + 6 × 8 3 10 + 8 ÷ 8

= 56 − . . . . . . . . . = . . . . . . . . . + . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

4 144 ÷ 12 × 2 5 6 × 3 + 5 × 8 6 72 − 64 ÷ 8

= . . . . . . . . . × 2 = . . . . . . . . . + 5 × 8 = . . . . . . . . . − . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . + . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

7 7 × 2 + 9 8 12 + 12 × 2 ÷ 6 9 6 + 6 × 4 + 8

= . . . . . . . . . + 9 = . . . . . . . . . + . . . . . . . . . ÷ 6 = . . . . . . . . . + . . . . . . . . . + . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . + . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

Find the value of each of the following, using the order of operations rules.a 7 + 8 ÷ 4 b 12 × 2 + 4 × 5

THINK WRITE

a Write the question. a 7 + 8 ÷ 4There are two operations to be performed: addition and division. Following the order of operations rules, perform division first, followed by addition.

= 7 + 2= 9

b Write the question. b 12 × 2 + 4 × 5There are three operations to be performed: multiplication, addition and another multiplication. According to the order of operations, multiplication must be done before the addition, from left to right. So, multiply 12 by 2 first, then multiply 4 by 5 and add the two products together.

= 24 + 4 × 5= 24 + 20= 44

12

12

WORKEDExample

92

SkillSHEETanswers10 11 × 8 + 9 × 3 11 5 × 4 − 2 × 9 12 20 − 8 × 2 ÷ 4

= . . . . . . . . . + 9 × . . . . . . . . . = . . . . . . . . . − . . . . . . . . . × 9 = . . . . . . . . . − . . . . . . . . . ÷ 4

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

93

SkillSHEETanswersSkillSHEET 5.3Order of operations with bracketsIf an expression contains more than one operation, the calculations must be performed in the followingorder:1. brackets (from left to right)2. multiplication and division (from left to right)3. addition and subtraction (from left to right).

Try theseFind the value of each of the following, using the order of operations rules.

1 2 × (15 − 6) 2 (10 − 8) × (4 + 2) 3 12 ÷ 4 × (54 − 49)

= 2 × . . . . . . . . = . . . . . . . . × (4 + 2) = . . . . . . . . ÷ . . . . . . . . × . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . × . . . . . . . . = . . . . . . . . . × . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

4 44 ÷ 11 × (16 − 5) 5 (18 + 6 × 8) × 2 6 5 × (63 − 9 × 6)

= . . . . . . . . ÷ . . . . . . . . . × . . . . . . . . = (18 + . . . . . . . . ) × . . . . . . . . = . . . . . . . . × (. . . . . . . . − . . . . . . . . )

= . . . . . . . . × . . . . . . . . = . . . . . . . . × . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

Find the value of the following, using the order of operations rules.a 5 × (2 + 4 × 7) b 4 × (3 × 5 + 5) × (7 − 4)

THINK WRITE

a Write the question. a 5 × (2 + 4 × 7)Expressions in brackets must be evaluated first. Inside the brackets there are two operations to be performed: addition and multiplication. According to the order of operations, multiplication must be done before the addition. Once the value of the expression in brackets is obtained, multiply it by 5.

= 5 × (2 + 28)= 5 × 30= 150

b Write the question. b 4 × (3 × 5 + 5) × (7 − 4)Expressions in brackets must be done first, followed by multiplication. Inside the first set of brackets, multiplication must be done before addition (which in this case is in the same order as it is written). So evaluate the contents of the first and second sets of brackets, and then multiply the three numbers together.

= 4 × (15 + 5) × (7 − 4)= 4 × 20 × (7 − 4)= 4 × 20 × 3= 80 × 3= 240

12

12

WORKEDExample

94

SkillSHEETanswers7 (4 + 21 ÷ 3) × 9 8 (6 + 3) × (7 + 2) 9 10 × (8 + 9 × 3)

= (. . . . . . . . ) × . . . . . . . . = . . . . . . . . × (7 + 2) = . . . . . . . . × (. . . . . . . . + . . . . . . . . )

= . . . . . . . . × . . . . . . . . = . . . . . . . . × . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

10 9 ÷ 3 × (4 + 3) 11 2 × 4 × (6 + 3 × 2) 12 (7 + 2) × (12 ÷ 4 − 3)

= . . . . . . . . ÷ . . . . . . . . × . . . . . . . . = . . . . . . . . × . . . . . . . . × (6. . . . . . . . ) = . . . . . . . . × (. . . . . . . . − . . . . . . . . )

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

95

SkillSHEETanswersSkillSHEET 5.4Operations with directed numbersFor addition of directed numbers follow these rules.1. If both numbers have the same sign, add them together; the answer will have the same sign. 2. If numbers have opposite signs, subtract the smaller number from the larger. The answer will have the

sign of the larger number. For subtraction of directed numbers, follow these rules.1. Replace subtraction with addition and change the sign of the number that is being subtracted to opposite.2. Perform the addition.For multiplication and division of directed numbers, follow these rules.1. Multiply (or divide) the numbers.2. If both numbers have the same sign, the product (quotient) will be positive.3. If two numbers have opposite signs, the product (quotient) will be negative.

Perform each of the following calculations.a -5 + -4 b 2 + -6 c -7 − -3 d 5 × -8 e -16 ÷ -4

THINK WRITE

a Since both numbers have the same sign (that is, they are both negative), add them together. The answer will have the same sign (negative) as the numbers in question.

a −5 + −4= −5 – 4= –9

b Since the numbers have different signs, subtract 2 from 6. The answer must have the sign of the larger number. So since 6 is larger than 2 and it is negative, the answer is also negative.

b 2 + −6= 2 – 6= –4

c To subtract negative 3 is the same as to add positive 3, so replace subtraction with addition and omit the negative sign in front of the 3.

c −7 − −3= −7 + 3

Since the numbers have different signs, subtract 3 from 7. Since 7 is larger than 3 and it is negative, the answer is also negative.

= −4

d Multiply 5 by 8. Since the numbers have opposite signs, the answer is negative.

d 5 × −8 = −40

e Divide 16 by 4. Since both numbers have the same sign, the answer is positive.

e −16 ÷ −4

=

= 4

1

2

16–4–

---------

WORKEDExample

96

SkillSHEETanswersTry thesePerform each of the following calculations.

1 −2 + −3 = . . . . . . . . . . . . . . . . . . . . . . . . 2 −2 + 5 = . . . . . . . . . . . . . . . . . . . . . . . .

3 −6 + 4 = . . . . . . . . . . . . . . . . . . . . . . . . 4 −12 + −4 = . . . . . . . . . . . . . . . . . . . . . . . .

5 −2 − −7 = . . . . . . . . . . . . . . . . . . . . . . . . 6 −8 − −12 = . . . . . . . . . . . . . . . . . . . . . . . .

7 5 − −7 = . . . . . . . . . . . . . . . . . . . . . . . . 8 −8 − 4 = . . . . . . . . . . . . . . . . . . . . . . . .

9 6 − −3 = . . . . . . . . . . . . . . . . . . . . . . . . 10 −6 − 5 = . . . . . . . . . . . . . . . . . . . . . . . .

11 −6 × −2 = . . . . . . . . . . . . . . . . . . . . . . . . 12 5 × −4 = . . . . . . . . . . . . . . . . . . . . . . . .

13 −8 × 7 = . . . . . . . . . . . . . . . . . . . . . . . . 14 12 ÷ −3 = . . . . . . . . . . . . . . . . . . . . . . . .

15 −20 ÷ −4 = . . . . . . . . . . . . . . . . . . . . . . . .

97

SkillSHEETanswersSkillSHEET 5.5Combining like termsTerms that contain exactly the same pronumeral(s) are called like terms. Like terms can be collectedtogether by adding (or subtracting) their coefficients. A coefficient is a number in front of the term. Notethat if there is no number, then the coefficient is 1.

Try theseSimplify each of the following expressions by combining like terms.

1 5x + 7x = ____x 2 11x − 7x = 4__

3 4x + 3x + 8x = . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6x − 2x − x = . . . . . . . . . . . . . . . . . . . . . . . . . .

5 9x − 5x + 2x = . . . . . . . . . . . . . . . . . . . . . . . . . . 6 x + 4x − 3x = . . . . . . . . . . . . . . . . . . . . . . . . . .

7 3x + 7x − 9 = . . . . . . . . . . . . . . . . . . . . . . . . . . 8 15x − 8 − 7x = . . . . . . . . . . . . . . . . . . . . . . . . . .

9 4x + 12 − x = . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2x + 5 + 3x − 2 = . . . . . . . . . . . . . . . . . . . . . . . . . .

11 7x + 8 − 2x + 7 = . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4x − 2x + 6 − x − 5 = . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplify each of the following expressions by combining like terms.a x + 3x - 2x b 3x + 6 + 4x - 1

THINK WRITE

a All three terms are like terms, as they contain the same pronumeral (x). Collect the three terms together by adding the coefficients of the first two terms and then subtracting the coefficient of the third term (1 + 3 − 2).

a x + 3x − 2x= 2x

b Collect the terms containing x by adding their coefficients together (3 + 4).

b 3x + 6 + 4x − 1= 7x + 6 − 1

Simplify further by subtracting 1 from 6. = 7x + 5

1

2

WORKEDExample

98

SkillSHEETanswersSkillSHEET 5.6Simplifying fractionsTo simplify a fraction, divide both numerator and denominator by the highest common factor.


1 = . . . . . . . . . . . . 2 = . . . . . . . . . . . . 3 = . . . . . . . . . . . .

4 = . . . . . . . . . . . . 5 = . . . . . . . . . . . . 6 = . . . . . . . . . . . .

7 = . . . . . . . . . . . . 8 = . . . . . . . . . . . . 9 = . . . . . . . . . . . .

10 = . . . . . . . . . . . . 11 = . . . . . . . . . . . . 12 = . . . . . . . . . . . .

13 = . . . . . . . . . . . . 14 = . . . . . . . . . . . . 15 = . . . . . . . . . . . .

16 = . . . . . . . . . . . . 17 = . . . . . . . . . . . . 18 = . . . . . . . . . . . .


THINK WRITE


=

852------

852------ 2

13------

WORKEDExample

1352------ 26

52------ 4

52------

1252------ 48

52------ 2

6---

46--- 8

10------ 15

20------

1220------ 8

26------ 24

42------

1848------ 7

56------ 12

36------

80360--------- 135

360--------- 99

360---------

99

SkillSHEETanswersSkillSHEET 5.7Highest common factorFactors that are the same for two or more numbers are called common factors. The largest of the commonfactors is called the highest common factor (HCF).

Try theseFind the highest common factor for the following pairs of numbers.

1 4 and 6 2 12 and 16

Factors of 4: 1, 2, 4 Factors of 12: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Factors of 6: 1, 2, ____, ____ Factors of 16: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Common factors: ____, ____ Common factors: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 9 and 27 4 10 and 18

Factors of 9: . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors of 10: . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Common factors: . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common factors: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 4 and 12 6 8 and 20




HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Find the highest common factor for the numbers 12 and 20.

THINK WRITE

List all factors of 12. 1, 2, 3, 4, 6, 12


Select the numbers that appear on both lists (that is, the common factors).

Common factors: 1, 2, 4

State the largest of the common factors. HCF = 4

1

2

3

4

WORKEDExample

100

SkillSHEETanswers7 15 and 18 8 16 and 24




HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 20 and 50 10 18 and 26




HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

SkillSHEETanswersSkillSHEET 6.1Flow chartsWhen constructing flow charts, we put numbers inside the boxes and operations on top of the arrows. Thenumber in the first box is called an input number, while the number in the last box is called an outputnumber. The output number is obtained by performing the operation, shown by the arrow, on the inputnumber.

Try these1 Find the output number in each of the following flow charts.

a b c d

e f g h

i j

Find the output number in the following flow chart.

THINK WRITE

Copy down the given flow chart.

The input number is 3; the operation shown by the arrow tells us that we need to add 2 to the input number. So the output number must be 5 because 3 + 2 = 5.

3

+ 2

1

3

+ 2

2

3 5

+ 2

1WORKEDExample

Complete the following flow chart by writing the missing number on top of the arrow.

THINK WRITE

Copy down the given flow chart.

The input number 4 is multiplied by a certain number to give 20. Since 4 × 5 = 20, the number on top of the arrow is 5.

4 20

×

1

4 20

×

2

4 20

× 5

2WORKEDExample

4

+ 3

6

– 4

6

× 2

8

÷ 4

5

+ 9

10

– 3

8

× 4

9

÷ 9

7

+ 8

11

– 7

102

SkillSHEETanswers2 Complete each of the following flow charts by writing the missing number on top of the arrow.

a b c d

e f g h

i j

Applying more than one operation

Try these3 Use 6 as your starting number, and find the result after applying the following operations.

a + 5, × 3 b − 2, ÷ 4

6 + 5 = . . . . . . . . . . . . . . . . . . . . . . . . . . − 2 = . . . . . . . . . . . . .

11 × . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . ÷ 4 = . . . . . . . . . . . . .

The result is . . . . . . . . . . . . . . . . . . . . . . . . . . The result is . . . . . . . . . . . . . . . . . . . . . . . . . .

c ÷ 2, + 10 d × 5, ÷ 3

. . . . . . . . . . . . . ÷ . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . + . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


8 12

+

7 2

–

10 50

×

16 2

÷

9 17

+

15 6

–

11 88

×

36 4

÷

19 29

+

17 12

–

Use 6 as your starting number, and find the result after applying the following operations.a + 4, ∏ 5 b ¥ 3, - 8

THINK WRITE

a Start with a 6 and perform the first operation (add 4).

a 6 + 4 = 10

Now start with the result of the first operation (10) and perform the second operation (divide by 5).

10 ÷ 5 = 2

Write down the result. The result is 2.

b Start with a 6 and perform the first operation (multiply by 3).

b 6 × 3 = 18

Now start with the result of the first operation (18) and perform the second operation (subtract 8).

18 − 8 = 10

Write down the result. The result is 10.

1

2

3

1

2

3

3WORKEDExample

103

SkillSHEETanswerse ÷ 3, − 2 f × 8, − 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


g − 5, + 18 h + 12, ÷ 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


i + 7, − 11 j ÷ 6, × 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


k − 4, × 11 l × 6, + 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


104

SkillSHEETanswersSkillSHEET 6.2Inverse operationsThe four basic operations and their inverse operations are shown in the table below:

Try theseWrite the inverse of each of the following.

1 + 5 . . . . . . . . . . . . . . . . . . . . . 2 − 7 . . . . . . . . . . . . . . . . . . . . . 3 × 6 . . . . . . . . . . . . . . . . . . . . . 4 ÷ 8 . . . . . . . . . . . . . . . . . . . . .

5 − 12 . . . . . . . . . . . . . . . . . . . . . 6 ÷ 9 . . . . . . . . . . . . . . . . . . . . . 7 × 11 . . . . . . . . . . . . . . . . . . . . . 8 + 8 . . . . . . . . . . . . . . . . . . . . .

9 − 10 . . . . . . . . . . . . . . . . . . . . . 10 ÷ 2 . . . . . . . . . . . . . . . . . . . . .

OperationInverse

operation

+ −

− +

× ÷

÷ ×

Write the inverse of each of the following.a ¥ 3 b - 4

THINK WRITE

a The inverse of multiplication is division; the number must remain unchanged. So the inverse of multiplying by 3 is dividing by 3.

a ÷ 3

b The inverse of subtraction is addition, so the inverse of minus 4 is plus 4.

b + 4

WORKEDExample

105

SkillSHEETanswersSkillSHEET 6.3Building expressionsTo build up an expression using x and the four operations, the information in the table below may be useful.

Note that in the above table the number 5 was used in all operations. It should be understood that any othernumber could be used in place of 5.

Try theseFill in the output column of the table with appropriate expressions.

Input Operation Output

x + 5 x + 5

x − 5 x − 5

x × 5 5x

x ÷ 5

Input Operation Output Input Operation Output

1 x + 4 2 x − 7

3 x × 8 4 x ÷ 4

5 3x + 2.6 6 5x − 17

7 x + 2 ÷ 9 8 x − 9 × 0.6

9 7x + 10 − 8

x5---

Fill in the output column with appropriate expressions.


a x - 12

b 3x ∏ 8

THINK WRITE

a We need to subtract 12 from the input number x. This is written as an algebraic expression.

a

b The input number 3x needs to be divided by 8. In algebra, division is usually denoted using

fractions. The output expression is .

b


x − 12 x − 12

3x8

------


3x ÷ 83x8

------

WORKEDExample

35---

x3---

106

SkillSHEETanswersSkillSHEET 6.4Solving equations by backtrackingTo solve equations using backtracking, construct a flow chart and then use inverse operations to find thevalue of the unknown. Remember that the inverse of addition is subtraction, the inverse of subtraction isaddition, the inverse of division is multiplication and the inverse of multiplication is division.

Solve the following equations, using backtracking.a x + 25 = 90 b 3x - 30 = 180

THINK WRITE

a Write the equation. a x + 25 = 90Construct a flow chart and build up an expression beginning with x.

Use backtracking and inverse operations to solve for x.

Write the solution. x = 65

b Write the equation. b 3x − 30 = 180Construct a flow chart and build up an expression beginning with x.

Use backtracking and inverse operations to find the value of x.

Write the solution. x = 70

12

+ 25

x

90

x + 25

3 + 25

– 25

x

9065

x + 25

4

12 × 3 − 30

x

180

3x 3x – 30

3 × 3 – 30

÷ 3 + 30

x

18021070

3x 3x – 30

4

WORKEDExample

107

SkillSHEETanswersTry theseSolve the following equations, using backtracking.

1 x + 55 = 90 2 x − 12 = 29

x = . . . . . . . . . . . . . . . . . . . . . x = . . . . . . . . . . . . . . . . . . . . .

3 3x = 51 4 = 14

x = . . . . . . . . . . . . . . . . . . . . . x = . . . . . . . . . . . . . . . . . . . . .

5 2x + 7 = 19 6 5x − 1 = 44

x = . . . . . . . . . . . . . . . . . . . . . x = . . . . . . . . . . . . . . . . . . . . .

7 4(x − 3) = 32 8 3(x + 7) = 30

x = . . . . . . . . . . . . . . . . . . . . . x = . . . . . . . . . . . . . . . . . . . . .

9 = 1 10 = 9

x = . . . . . . . . . . . . . . . . . . . . . x = . . . . . . . . . . . . . . . . . . . . .

x5---

x 4–5

----------- x 6+4

------------

108

SkillSHEETanswersSkillSHEET 6.5Combining like termsTerms that contain exactly the same pronumeral(s) are called like terms. Like terms can be collectedtogether by adding (or subtracting) their coefficients. A coefficient is a number in front of the term. Notethat if there is no number, then the coefficient is 1.

Try theseSimplify each of the following expressions by combining like terms.

1 5x + 7x = ____x 2 11x − 7x = 4__

3 4x + 3x + 8x = . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6x − 2x − x = . . . . . . . . . . . . . . . . . . . . . . . . . .

5 9x − 5x + 2x = . . . . . . . . . . . . . . . . . . . . . . . . . . 6 x + 4x − 3x = . . . . . . . . . . . . . . . . . . . . . . . . . .

7 3x + 7x − 9 = . . . . . . . . . . . . . . . . . . . . . . . . . . 8 15x − 8 − 7x = . . . . . . . . . . . . . . . . . . . . . . . . . .

9 4x + 12 − x = . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2x + 5 + 3x − 2 = . . . . . . . . . . . . . . . . . . . . . . . . . .

11 7x + 8 − 2x + 7 = . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4x − 2x + 6 − x − 5 = . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplify each of the following expressions by combining like terms.a x + 3x - 2x b 3x + 6 + 4x - 1

THINK WRITE

a All three terms are like terms, as they contain the same pronumeral (x). Collect the three terms together by adding the coefficients of the first two terms and then subtracting the coefficient of the third term (1 + 3 − 2).

a x + 3x − 2x= 2x

b Collect the terms containing x by adding their coefficients together (3 + 4).

b 3x + 6 + 4x − 1= 7x + 6 − 1

Simplify further by subtracting 1 from 6. = 7x + 5

1

2

WORKEDExample

109

SkillSHEETanswersSkillSHEET 6.6Expanding expressions containing bracketsTo expand (multiply out) expressions containing brackets, multiply each term in the brackets by thecoefficient in front.

Try theseExpand each of the following.

1 3(2x + 1) 2 2(4x + 2)

= 3 × . . . . . . . + 3 × . . . . . . . = . . . . . . . × 4x + . . . . . . . × . . . . . . .

= . . . . . . . + . . . . . . . = . . . . . . . + . . . . . . .

3 7(x − 5) 4 6(5x − 4)

= . . . . . . . × . . . . . . . + . . . . . . . × −5 = . . . . . . . × . . . . . . . + . . . . . . . × . . . . . . .

= . . . . . . . − . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . .

5 4(3x − 1) 6 − 2(2x + 3)

= . . . . . . . . . . . . . . . . . . . . . . . . . . = −2 × . . . . . . . + . . . . . . . × . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . .

7 − 3(3x + 4) 8 − 4(2x − 2)

= . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . × . . . . . . . + . . . . . . . × . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . .

9 − 5(x − 2) 10 − 6(x − 1)

= . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . .

Expand each of the following.a 3(2x − 4) b -2(4x + 7)

THINK WRITE

a Write the question. a 3(2x − 4)Multiply 3 by 2x and then 3 by − 4. = 3 × 2x + 3 × − 4Simplify. = 6x − 12

b Write the question. b −2(4x + 7)Multiply −2 by 4x and then −2 by 7. = −2 × 4x + − 2 × 7Simplify. = − 8x − 14

123

123

WORKEDExample

110

SkillSHEETanswersSkillSHEET 6.7Checking solutions by substitutionA solution to an equation is a number that when substituted into the equation makes it a true statement.

To check whether the given number is the solution to a particular equation, replace the pronumeral withthe given number and see if the left side of the equation equals the right side. If it is, then the given numberis the solution to the equation.

Try theseAnswer true or false to each of the following.

1 The solution to the equation x + 10 = 24 is 14.

Left side = x + . . . . . . . . . . . . . Right side = 24

Left side = . . . . . . . . . . . . . + . . . . . . . . . . . . .

Left side = . . . . . . . . . . . . .

The statement is . . . . . . . . . . . . . . . . . . . . . . . . . .

2 The solution to the equation x − 8 = 11 is 20.

Left side = x − 8 Right side = . . . . . . . . . . . . .

Left side = . . . . . . . . . . . . . − 8

Left side = . . . . . . . . . . . . .

The statement is . . . . . . . . . . . . . . . . . . . . . . . . . .

Answer true or false to each of the following.a The solution to the equation x + 4 = 16 is 12.b The solution to the equation 6x − 5 = x + 6 is 3.

THINK WRITE

a Copy the equation, then consider the left and right sides.

a x + 4 = 16

Replace x with 12 (the number suggested as a solution).

Left side = x + 4 Right side = 16

Simplify the left side (12 + 4 = 16). Left side = 12 + 4Left side = 16

Consider the result (16 = 16): it is a true statement, so state your conclusion.

The statement is true.

b Copy the equation, then consider the left and right sides.

b 6x − 5 = x + 6Left side = 6x − 5 Right side = x + 6

Replace x with 3 (the number suggested as a solution).Simplify each side of the equation.

= 6 × 3 − 5 = 3 + 6= 18 − 5 = 9= 13

The result (13 = 9) is obviously not true, so state your conclusion.

The statement is false.

1

2

3

4

1

2

34

WORKEDExample

111

SkillSHEETanswers3 The solution to the equation 6x = 42 is 8.

Left side = 6x Right side = 42

Left side = 6 × . . . . . . . . . . . . .

Left side = . . . . . . . . . . . . .

The statement is . . . . . . . . . . . . . . . . . . . . . . . . . .

4 The solution to the equation = 5x is 8.

Left side = Right side = . . . . . . . . . . . . .

Left side = . . . . . . . . . . . . . ÷ . . . . . . . . . . . . . Right side = 5 × . . . . . . . . . . . . .

Left side = . . . . . . . . . . . . . Right side = . . . . . . . . . . . . .

The statement is . . . . . . . . . . . . . . . . . . . . . . . . . .

5 The solution to the equation + 7 = 12 is 10.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The statement is . . . . . . . . . . . . . . . . . . . . . . . . . .

6 The solution to the equation 8x − 4 = 5x + 5 is 3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The statement is . . . . . . . . . . . . . . . . . . . . . . . . . .

7 The solution to the equation 11x + 8 = 12x + 13 is 6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The statement is . . . . . . . . . . . . . . . . . . . . . . . . . .

x5---

x5---

x2---

112

SkillSHEETanswersSkillSHEET 6.8Writing equations from worded statementsFollow these steps to write an equation from a worded sentence.Step 1 Break the sentence into parts.Step 2 Identify what type of algebra or arithmetic is required.Step 3 Connect individual parts to form an equation.

The table below shows some common words and expressions and their mathematical meaning.

We also need to remember the following rules.1. The multiplication sign between the number and the pronumeral is omitted (not written). For example,

2 × x is written as 2x.2. Division of algebraic terms is usually written as a fraction with the number that is being divided (the

dividend) in the numerator and the number by which we are dividing (the divisor) in the denominator.

For example, x ÷ 2 is written as .

Word expression Operation

‘the sum of’, ‘is added to’, ‘plus’ +

‘the difference between’, ‘is subtracted from’, ‘minus’ –

‘the product of’, ‘is multiplied by’, ‘times’ ×

‘the quotient of’, ‘is divided by’ ÷

‘is equal to’, ‘the result is’, ‘gives’ =

x2---

113

SkillSHEETanswers

Try theseWrite an equation for each of the following statements, using x to represent the unknown number.

1 When 3 is added to a certain number, the result is 100.

‘3 is added to’ means . . . . . . . . . . . . . .

‘A certain number’ means . . . . . . . . . . . . . .

‘The result is 100’ means . . . . . . . . . . . . . .

Equation: x + . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

2 Nine times a certain number is 72.

‘Nine times’ means . . . . . . . . . . . . . .


‘Is 72’ means . . . . . . . . . . . . . .

Equation: . . . . . . . . . . . . . . = 72

3 When 4 is subtracted from a certain number, the result is 19.

‘4 is subtracted from’ means . . . . . . . . . . . . . .



Equation: . . . . . . . . . . . . . .

4 Dividing a certain number by 5 gives 12.

‘Dividing by’ means . . . . . . . . . . . . . .


‘Gives 12’ means . . . . . . . . . . . . . .

Equation: . . . . . . . . . . . . . .

Write an equation for each of the following statements, using x to represent the unknown number.a Five times a certain number gives 15.b When 12 is subtracted from a certain number, the result is 6.

THINK WRITE

a Break the sentence into parts and identify the type of arithmetic or algebra required.

a ‘Five times’ means ‘5 ×’‘A certain number’ means x‘Gives 15’ means ‘= 15’

Form an equation by combining the individual parts together.

5 × x = 155x = 15

b Break the sentence into parts and identify the type of arithmetic or algebra required.

b ‘12 is subtracted from’ means ‘– 12’‘A certain number’ means x‘The result is 6’ means ‘= 6’

Form an algebraic equation from the individual parts. Note that the number from which we are subtracting is written first, while the number that is being subtracted is written second. In this case, x is written first and 12 second, since 12 is subtracted from a certain number.

x –12 = 6

1

2

1

2

WORKEDExample

114

SkillSHEETanswers5 The difference between a certain number and 7 is 3.

‘The difference between’ means . . . . . . . . . . . . . .


‘Is 3’ means . . . . . . . . . . . . . .

Equation: . . . . . . . . . . . . . .

6 The product of a certain number and 12 is 88.

‘The product of’ means . . . . . . . . . . . . . .


‘Is 88’ means . . . . . . . . . . . . . .

Equation . . . . . . . . . . . . . .

7 The sum of 23 and a certain number is 50.

‘The sum of’ means . . . . . . . . . . . . . .


‘Is 50’ means . . . . . . . . . . . . . .

Equation . . . . . . . . . . . . . .

8 When a certain number is divided by 10, the result is 3.


‘Is divided by’ means . . . . . . . . . . . . . .


Equation . . . . . . . . . . . . . .

9 Seven times a certain number gives 56.

‘Seven times’ means . . . . . . . . . . . . . .


‘Gives 56’ means . . . . . . . . . . . . . .

Equation . . . . . . . . . . . . . .

10 Five more than a certain number is 9.

‘Five more than’ means . . . . . . . . . . . . . .


‘Is 9’ means . . . . . . . . . . . . . .

Equation . . . . . . . . . . . . . .

115

SkillSHEETanswersSkillSHEET 6.9Checking solutions to inequationsA given value is a possible solution to an inequation (or inequality) if, when substituted in place of the pro-numeral, the value makes the inequality a true statement.

Follow these steps to determine whether a given value is a possible solution to the inequality.Step 1 Substitute the given value for the pronumeral in the inequation and simplify.Step 2 Consider the result: if it is a true statement, the given value is a possible solution to the inequation;

if the statement is false, the given value is not a solution.

Try theseState whether the value given in brackets is a possible solution to the inequation for each of the following.

1 3x < 18 [x = 6] 2 ≥ 2 [b = 8]

Is 3 × 6 < . . . . . . . . . . . . . .? Is ≥ . . . . . . . . . . . . . .?

Is 18 < . . . . . . . . . . . . . .? . . . . . . . . . . . . . . Is . . . . . . . . . . . . . . ≥ . . . . . . . . . . . . . .? . . . . . . . . . . . . . .

So x = 6 . . . . . . . . . . . . . . a possible solution So b = 8 . . . . . . . . . . . . . . a possible solutionto the given inequation. to the given inequation.

State whether the value given in brackets is a possible solution to the inequation for each of the following.

a 5x < 16 [x = 3] b ≥ –2 [a = –9]

THINK WRITE

a Write the inequation. a 5x < 16Substitute 3 for x in the inequation. Is 5 × 3 < 16?Simplify the left-hand side and consider whether the statement is true.

Is 15 < 16? Yes

State your conclusion. So x = 3 is a possible solution to the given inequation.

b Write the inequation. b ≥ –2

Substitute –9 for a in the inequation.Is ≥ –2?

Simplify the left-hand side and consider whether the statement is true.

Is –3 ≥ –2? No

State your conclusion. So a = –9 is not a possible solution to the given inequation.

a3---

123

4

1a3---

2 9–3

------

3

4

WORKEDExample

b4---

8

.......--------

116

SkillSHEETanswers3 –4y > 16 [y = –5] 4 5p ≤ 1 [p = –2]

Is –4 × . . . . . . . . . . . . . . > . . . . . . . . . . . . . .? Is . . . . . . . . . . . . . . × . . . . . . . . . . . . . . ≤ . . . . . . . . . . . . . .?

Is . . . . . . . . . . . . . . > . . . . . . . . . . . . . .? . . . . . . . . . . . . . . Is . . . . . . . . . . . . . . ≤ . . . . . . . . . . . . . .? . . . . . . . . . . . . . .

So y = –5 . . . . . . . . . . . . . . a possible solution So p = –2 . . . . . . . . . . . . . . a possible solutionto the given inequation. to the given inequation.

5 ≤ –3 [c = –14] 6 12x < 30 [x = 3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

So c = –14 . . . . . . . . . . . . . . a possible solution So x = 3 . . . . . . . . . . . . . . a possible solutionto the given inequation. to the given inequation.

7 > –5 [d = 8] 8 5u ≥ –16 [u = –3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

So d = 8 . . . . . . . . . . . . . . a possible solution So u = –3 . . . . . . . . . . . . . . a possible solutionto the given inequation. to the given inequation.

9 > 4 [a = 20] 10 k + 2 ≤ 5 [k = 3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

So a = 20 . . . . . . . . . . . . . . a possible solution So k = 3 . . . . . . . . . . . . . . a possible solutionto the given inequation. to the given inequation.

c7---

d2–

------

a5---

117

SkillSHEETanswersSkillSHEET 6.10Showing inequations on a number lineInequations of the type x > a, x < a, x ≥ a and x ≤ a (where a is any number) can be represented using anumber line, by following these steps:Step 1 Locate the required number a on the number line.Step 2 If the sign of inequality is < or >, put an open or empty circle over number a (to indicate that this

number is not included). If the sign of inequality is ≥ or ≤, place a closed or solid circle abovenumber a (to indicate that the number is included).

Step 3 If we need to show numbers greater than (>) or greater than or equal to (≥) a, draw an arrow to theright of the circle (as numbers on the line increase from left to right). If we need to show numbersless than (<) or less than or equal to (≤) a, draw an arrow to the left of the circle.

The four basic inequalities and their graphical representation are shown in the table below. Note that in theexamples shown the value of a is 1.

Mathematical statement Worded statement Number line diagram

x > 1 x is greater than 1

x ≥ 1 x is greater than or equal to 1

x < 1 x is less than 1

x ≤ 1 x is less than or equal to 1

–1 0 1 2–2–3–4 3 4

–1 0 1 2–2–3–4 3 4

–1 0 1 2–2–3–4 3 4

–1 0 1 2–2–3–4 3 4

Indicate each of the following on a number line.a x > –2 b x £ 3

THINK WRITE

a Draw a number line and locate −2. Since the sign of inequality is >, place an empty circle above −2 (to indicate that −2 is not included) and draw an arrow to the right of the circle.

a

b Draw a number line and locate 3. Since the sign of inequality is ≤, place a solid circle above 3 (to indicate that 3 itself is included) and draw an arrow to the left of the circle.

b

–1 0 1 2–2–3–4 3 4

–1 0 1 2–2–3–4 3 4

WORKEDExample

118

SkillSHEETanswersTry theseIndicate each of the following on the number lines.

1 x > 4 2 x ≤ – 1

3 x ≥ 0 4 x < 2

5 x ≤ 4 6 x > –2

7 x < –3 8 x ≥ 2

–1 0 1 2–2–3–4 3 4 –1 0 1 2–2–3–4 3 4

–1 0 1 2–2–3–4 3 4 –1 0 1 2–2–3–4 3 4

–1 0 1 2–2–3–4 3 4 –1 0 1 2–2–3–4 3 4

–1 0 1 2–2–3–4 3 4 –1 0 1 2–2–3–4 3 4

119

SkillSHEETanswersSkillSHEET 7.1Converting units of length, capacity and timeConverting units of lengthTo convert units of length, the following chart may be useful.

Try these1 Convert each of the following to the units given in brackets.

a 70 m (cm) b 350 m (cm) c 5 m (cm)

70 × . . . . . . . . . = . . . . . . . . . 350 × . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . .

70 m = . . . . . . . . . cm 350 m = . . . . . . . . . cm 5 m = . . . . . . . . . cm

d 10 km (cm) e 400 km (cm) f 5000 km (m)

10 × . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 km = . . . . . . . . . cm .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g 200 cm (m) h 500 mm (cm) i 8000 m (km)

200 ÷ . . . . . . . . . = . . . . . . . . . 500 ÷ . . . . . . . . . = . . . . . . . . . . . . . . . . . . ÷ . . . . . . . . . = . . . . . . . . .

200 cm = . . . . . . . . . m 500 mm = . . . . . . . . . cm 8000 m = . . . . . . . . . km

mm cm

÷ 10 ÷ 100 ÷ 1000

× 10 × 100 × 1000

m km

Convert each of the following to the units given in brackets.a 20 m (cm) b 3000 m (km)

THINK WRITE

a To convert to smaller units, we need to multiply. Since 1 m = 100 cm, multiply by 100.

a 20 × 100 = 2000

Write the answer. 20 m = 2000 cm

b To convert to larger units, we need to divide. Since 1 km = 1000 m, divide by 1000.

b 3000 ÷ 1000 = 3

Write the answer. 3000 m = 3 km

1

2

1

2

1WORKEDExample

120

SkillSHEETanswersj 200 000 cm (km) k 50 cm (m) l 750 m (km)

200 000 ÷ . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200 000 cm = . . . . . . . . . km .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

m 78 m (cm) n 25 km (cm) o 8 m (cm)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Converting units of capacityThe following chart can be used to convert between kilolitres, litres and millilitres.

The following conversions may also come in handy.

1 cm3 = 1 mL 1 000 000 cm3 = 1 m3

1000 cm3 = 1 L 1 m3 = 1 kL

Try these2 Convert each of the following to the units in grouping symbols.

a 2 L (mL) b 3000 mL (L)

2 L = 2 × . . . . . . . . . mL 3000 mL = 3000 ÷ . . . . . . . . . L

2 L = . . . . . . . . . mL 3000 mL = . . . . . . . . . L

kL L mL

×1000 ×1000

÷1000 ÷1000

Copy and complete the following unit conversions.a 6 L = mL b 700 mL = L c 0.45 L = cm3

THINK WRITE

a Check the conversion chart.To convert litres to millilitres, multiply by 1000.

a 6 L = 6 × 1000 mL= 6000 mL

b Check the conversion chart.To convert millilitres to litres, divide by 1000.

b 700 mL = 700 ÷ 1000 L= 0.7 L

c Check the conversion chart.To convert litres to millilitres, multiply by 1000. Note that 1 mL = 1 cm3.

c 0.45 L = 0.45 × 1000 mL= 450 mL= 450 cm3

2WORKEDExample

121

SkillSHEETanswersc 13 kL (L) d 7000 mL (L)

13 kL = . . . . . . . . . × . . . . . . . . . L 7000 mL = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

13 kL = . . . . . . . . . L 7000 mL = . . . . . . . . . L

e 5500 mL (L) f 260 L (kL)

5500 mL = . . . . . . . . . . . . . . . . . . L 260 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . kL

5500 mL = . . . . . . . . . . . . . . . . . . L 260 L = . . . . . . . . . . . . . . . . . . kL

g 2.5 L (mL) h 32 000 mL (L)

2.5 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . mL 32 000 mL = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

2.5 L = . . . . . . . . . mL 32 000 mL = . . . . . . . . . L

i 55 mL (L) j 0.035 L (mL)

55 mL = . . . . . . . . . . . . . . . . . . . . . . . . . . . mL 0.035 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

55 mL = . . . . . . . . . mL 0.035 L = . . . . . . . . . L

k 420 L (mL) l 0.99 kL (L)

420 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . mL 0.99 kL = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

420 L = . . . . . . . . . mL 0.99 kL = . . . . . . . . . L

Converting units of timeUnits of time can be converted as follows.1 year = 12 months = 26 fortnights = 52 weeks 1 hour = 60 minutes1 fortnight = 2 weeks 1 minute = 60 seconds

Convert each of the following to the units shown in brackets.a 2 years (months) b 3 fortnights (weeks)

THINK WRITE

a One year = 12 months, so to convert years to months multiply the number of years by 12.

a 2 years = 2 × 12 months= 24 months

b One fortnight = 2 weeks, so to convert fortnights to weeks, multiply the number of fortnights by 2.

b 3 fortnights = 3 × 2 weeks= 6 weeks

3WORKEDExample

122

SkillSHEETanswersTry these3 Convert each of the following to the units shown in brackets.

a 3 years (months) b 2 years (weeks)

3 years = 3 × . . . . . . . . . months 2 years = . . . . . . . . . × . . . . . . . . . weeks

3 years = . . . . . . . . . months 2 years = . . . . . . . . . weeks

c 4 years (fortnights) d 5 fortnights (weeks)

4 years = . . . . . . . . . × . . . . . . . . . fortnights 5 fortnights = . . . . . . . . . × . . . . . . . . . weeks

4 years = . . . . . . . . . fortnights 5 fortnights = . . . . . . . . . weeks

e 48 months (years) f 24 weeks (fortnights)

48 months = 48 ÷ . . . . . . . . . years 24 weeks = . . . . . . . . . ÷ . . . . . . . . . fortnights

48 months = . . . . . . . . . years 24 weeks = . . . . . . . . . fortnights

g 18 months (years) h 5 years (months)

18 months = . . . . . . . . . . . . . . . . . . years 5 years = . . . . . . . . . . . . . . . . . . months

18 months = . . . . . . . . . years 5 years = . . . . . . . . . months

i 4 fortnights (weeks) j 5 hours (minutes)

4 fortnights = . . . . . . . . . . . . . . . . . . weeks 5 hours = . . . . . . . . . . . . . . . . . . minutes

4 fortnights = . . . . . . . . . weeks 5 hours = . . . . . . . . . minutes

k 300 seconds (minutes) l 3 hours (seconds)

300 seconds = . . . . . . . . . . . . . . . . . . minutes 3 hours = 3 × . . . . . . . . . minutes

300 seconds = . . . . . . . . . minutes 3 hours = . . . . . . . . . minutes

300 seconds = ?? minutes 3 hours = . . . . . . . . . × . . . . . . . . . seconds

3 hours = . . . . . . . . . seconds

123

SkillSHEETanswersSkillSHEET 7.2Highest common factorFactors that are the same for two or more numbers are called common factors. The largest of the commonfactors is called the highest common factor (HCF).

Try theseFind the highest common factor for the following pairs of numbers.

1 4 and 6 2 12 and 16

Factors of 4: 1, 2, 4 Factors of 12: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Factors of 6: 1, 2, ____, ____ Factors of 16: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Common factors: ____, ____ Common factors: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 9 and 27 4 10 and 18




HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 4 and 12 6 8 and 20




HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Find the highest common factor for the numbers 12 and 20.

THINK WRITE



Select the numbers that appear on both lists (that is, the common factors).

Common factors: 1, 2, 4

State the largest of the common factors. HCF = 4

1

2

3

4

WORKEDExample

124

SkillSHEETanswers7 15 and 18 8 16 and 24




HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 20 and 50 10 18 and 26




HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . . HCF: . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

SkillSHEETanswersSkillSHEET 7.3Simplifying fractionsTo simplify a fraction, divide both numerator and denominator by the highest common factor (HCF).


1 = . . . . . . . . . . . . 2 = . . . . . . . . . . . . 3 = . . . . . . . . . . . .

4 = . . . . . . . . . . . . 5 = . . . . . . . . . . . . 6 = . . . . . . . . . . . .

7 = . . . . . . . . . . . . 8 = . . . . . . . . . . . . 9 = . . . . . . . . . . . .

10 = . . . . . . . . . . . . 11 = . . . . . . . . . . . . 12 = . . . . . . . . . . . .

13 = . . . . . . . . . . . . 14 = . . . . . . . . . . . . 15 = . . . . . . . . . . . .

16 = . . . . . . . . . . . . 17 = . . . . . . . . . . . . 18 = . . . . . . . . . . . .


THINK WRITE


=

852------

852------ 2

13------

WORKEDExample

1352------ 26

52------ 4

52------

1252------ 48

52------ 2

6---

46--- 8

10------ 15

20------

1220------ 8

26------ 24

42------

1848------ 7

56------ 12

36------

80360--------- 135

360--------- 99

360---------

126

SkillSHEETanswersSkillSHEET 7.4Finding and converting to the lowest common denominatorTo obtain the lowest common denominator (LCD) of two or more fractions, we need to find the lowestcommon multiple (LCM) of the denominators of these fractions.

To convert a fraction to the LCD, we need to establish how many times the original denominator fits ordivides into the LCD. We then need to multiply both numerator and denominator of the fraction by thatnumber.

Try these1 Find the lowest common denominator of each of the following pairs of fractions.

a and b and

Multiples of 7: . . . . . . . . . . . . Multiples of 4: . . . . . . . . . . . .


LCM = . . . . . . . . . . . . LCM = . . . . . . . . . . . .

LCD of and is . . . . . . . . . . . . LCD of and is . . . . . . . . . . . .

Find the lowest common denominator of and .

THINK WRITE

To obtain the lowest common denominator of the given fractions, we need to find the LCM of 6 and 8. To find the LCM, list some multiples of 6 and 8 and select the smallest number that is on both lists.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64LCM = 24

LCM represents the lowest common denominator, so write the answer.

LCD of and is 24.

56--- 3

8---

1

256--- 3

8---

1WORKEDExample

Convert and to fractions with the lowest common denominator of 24.

THINK WRITE

Consider . The denominator 6 goes into 24 (the LCD) 4 times. So multiply both numerator and denominator of by 4.

= =

Consider . The denominator 8 goes into 24 (the LCD) 3 times. So multiply both numerator and denominator of by 3.

= =

56--- 3

8---

156---

56---

56--- 5 4×

6 4×------------ 20

24------

238---

38---

38--- 3 3×

8 3×------------ 9

24------

2WORKEDExample

57--- 1

6--- 3

4--- 2

3---

57--- 1

6--- 3

4--- 2

3---

127

SkillSHEETanswersc and d and



LCM = . . . . . . . . . . . . LCM = . . . . . . . . . . . .


e and f and



LCM = . . . . . . . . . . . . LCM = . . . . . . . . . . . .


g and h and



LCM = . . . . . . . . . . . . LCM = . . . . . . . . . . . .


2 Convert each pair of fractions in question 1 to fractions with their respective lowest commondenominators.

a = = b = =

= = = =

c = = d = =

= = = =

e = = f = =

= = = =

16--- 1

4--- 2

3--- 3

5---

16--- 1

4--- 2

3--- 3

5---

23--- 5

6--- 1

6--- 5

8---

23--- 5

6--- 1

6--- 5

8---

38--- 3

4--- 5

9--- 5

6---

38--- 3

4--- 5

9--- 5

6---

57--- 5 .......×

7 .......×----------------- .......

.......-------- 3

4--- 3

4--- .......

.......--------

16--- 1 .......×

6 .......×----------------- .......

.......-------- 2

3--- 2 .......×

3 .......×----------------- .......

.......--------

16--- ....... .......×

....... .......×----------------------- .......

.......-------- 2

3--- ....... .......×

....... .......×----------------------- .......

.......--------

14--- ....... .......×

....... .......×----------------------- .......

.......-------- 3

5--- ....... .......×

....... .......×----------------------- .......

.......--------

23--- ....... .......×

....... .......×----------------------- .......

.......-------- 1

6--- ....... .......×

....... .......×----------------------- .......

.......--------

56--- ....... .......×

....... .......×----------------------- .......

.......-------- 5

8--- ....... .......×

....... .......×----------------------- .......

.......--------

128

SkillSHEETanswersg = = h = =

= = = =

38--- ....... .......×

....... .......×----------------------- .......

.......-------- 5

9--- ....... .......×

....... .......×----------------------- .......

.......--------

34--- ....... .......×

....... .......×----------------------- .......

.......-------- 5

6--- ....... .......×

....... .......×----------------------- .......

.......--------

129

SkillSHEETanswersSkillSHEET 7.5Converting a mixed number to an improper fractionTo convert a mixed number into an improper fraction, follow these steps.1. Multiply the denominator by the whole part and add the numerator. 2. Put the resultant number as the numerator of the improper fraction. 3. Write the denominator. (It is the same as the one in the mixed number.)


1 1 = 2 2 = 3 1 = . . . . . . . . . . . .

4 2 = . . . . . . . . . . . .

5 3 = . . . . . . . . . . . .

6 2 = . . . . . . . . . . . .

7 4 = . . . . . . . . . . . .

8 1 = . . . . . . . . . . . .

9 5 = . . . . . . . . . . . .

10 4 = . . . . . . . . . . . .


THINK WRITE


2 =

38---

38--- 19

8------

WORKEDExample

35---

. . .

5---- 5

8---

21

...------ 2

7---

78--- 1

2--- 4

5---

34--- 1

9--- 5

6---

37---

130

SkillSHEETanswersSkillSHEET 7.6Multiplying decimals by 10, 100 or 1000To multiply a decimal number by a power of 10, move the decimal point to the right one place for each zeroin the power of 10. For example, to multiply by 10 move the decimal point one place to the right, and tomultiply by 1000 move it three places to the right. Note that if there are not enough digits after the decimalpoint, we can always add extra zeros.

Try these


1 2.56 × 10 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7.6 × 10 = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 0.98 × 10 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.49 × 100 = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 2.6 × 100 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 70.1 × 100 = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 0.2 × 100 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.321 × 1000 = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10.2 × 1000 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 0.758 × 1000 = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 2.5 × 100 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.576 × 10 = . . . . . . . . . . . . . . . . . . . . . . . . . . . .


THINK WRITE

a To multiply a decimal number by 10, move the decimal point one place to the right (as there is one zero in 10).

a 5.67 × 10 = 56.7

b To multiply a decimal number by 100, we need to move the decimal point two places to the right. However, there is only one digit after the decimal point (7). So add a zero first (to create two decimal places), and then move the decimal point. Note that we write the answer as 70, rather than 070.

b 0.7 × 100= 0.70 × 100= 70

WORKEDExample

131

SkillSHEETanswersSkillSHEET 7.7Multiplying a whole number by a fractionTo multiply a whole number by a fraction:1. change the whole number into a fraction by writing it over 12. simplify as much as possible3. multiply the numerators together and the denominators together4. if the answer is an improper fraction, convert it to a mixed number.

Try thesePerform each of the following multiplications.

1 × 50 2 × 27 3 × 40

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Perform each of the following multiplications.a × 50 b 75 ×

THINK WRITE

a Write the question. a × 50


= ×


= ×


=


= 30

b Write the question. b 75 ×


= ×


= ×


=


= 100

35--- 4

3---

135---

235--- 50

1------

331--- 10

1------

4 301------

5

143---

2751------ 4

3---

3251------ 4

1---

4 1001

---------

5

WORKEDExample

25--- 1

3--- 3

10------

25---

.......

.......-------- .......

.......-------- 27

.......-------- .......

.......-------- .......

.......--------

2

.......-------- .......

1--------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

132

SkillSHEETanswers4 16 × 5 60 × 6 81 ×

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 42 × 8 36 × 9 12 ×

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 × 14 11 × 200 12 × 35

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 28 × 14 20 × 15 15 ×

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34--- 5

12------ 2

9---

16

.......-------- 3

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

78--- 4

15------ 4

3---

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

107------ 23

10------ 6

5---

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

94--- 7

6--- 3

2---

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

133

SkillSHEETanswersSkillSHEET 7.8Converting units of length, capacity and timeConverting units of lengthTo convert units of length, the following chart may be useful.

Try these1 Convert each of the following to the units given in brackets.

a 70 m (cm) b 350 m (cm) c 5 m (cm)

70 × . . . . . . . . . = . . . . . . . . . 350 × . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . .

70 m = . . . . . . . . . cm 350 m = . . . . . . . . . cm 5 m = . . . . . . . . . cm

d 10 km (cm) e 400 km (cm) f 5000 km (m)

10 × . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 km = . . . . . . . . . cm .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g 200 cm (m) h 500 mm (cm) i 8000 m (km)

200 ÷ . . . . . . . . . = . . . . . . . . . 500 ÷ . . . . . . . . . = . . . . . . . . . . . . . . . . . . ÷ . . . . . . . . . = . . . . . . . . .

200 cm = . . . . . . . . . m 500 mm = . . . . . . . . . cm 8000 m = . . . . . . . . . km

mm cm

÷ 10 ÷ 100 ÷ 1000

× 10 × 100 × 1000

m km

Convert each of the following to the units given in brackets.a 20 m (cm) b 3000 m (km)

THINK WRITE


a 20 × 100 = 2000



b 3000 ÷ 1000 = 3


1

2

1

2

1WORKEDExample

134

SkillSHEETanswersj 200 000 cm (km) k 50 cm (m) l 750 m (km)

200 000 ÷ . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200 000 cm = . . . . . . . . . km .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

m 78 m (cm) n 25 km (cm) o 8 m (cm)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Converting units of capacityThe following chart can be used to convert between kilolitres, litres and millilitres.

The following conversions may also come in handy.

1 cm3 = 1 mL 1 000 000 cm3 = 1 m3

1000 cm3 = 1 L 1 m3 = 1 kL

Try these2 Convert each of the following to the units in grouping symbols.

a 2 L (mL) b 3000 mL (L)

2 L = 2 × . . . . . . . . . mL 3000 mL = 3000 ÷ . . . . . . . . . L

2 L = . . . . . . . . . mL 3000 mL = . . . . . . . . . L

kL L mL

×1000 ×1000

÷1000 ÷1000

Copy and complete the following unit conversions.a 6 L = mL b 700 mL = L c 0.45 L = cm3

THINK WRITE

a Check the conversion chart.To convert litres to millilitres, multiply by 1000.

a 6 L = 6 × 1000 mL= 6000 mL

b Check the conversion chart.To convert millilitres to litres, divide by 1000.

b 700 mL = 700 ÷ 1000 L= 0.7 L

c Check the conversion chart.To convert litres to millilitres, multiply by 1000. Note that 1 mL = 1 cm3.

c 0.45 L = 0.45 × 1000 mL= 450 mL= 450 cm3

2WORKEDExample

135

SkillSHEETanswersc 13 kL (L) d 7000 mL (L)

13 kL = . . . . . . . . . × . . . . . . . . . L 7000 mL = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

13 kL = . . . . . . . . . L 7000 mL = . . . . . . . . . L

e 5500 mL (L) f 260 L (kL)

5500 mL = . . . . . . . . . . . . . . . . . . L 260 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . kL

5500 mL = . . . . . . . . . . . . . . . . . . L 260 L = . . . . . . . . . . . . . . . . . . kL

g 2.5 L (mL) h 32 000 mL (L)

2.5 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . mL 32 000 mL = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

2.5 L = . . . . . . . . . mL 32 000 mL = . . . . . . . . . L

i 55 mL (L) j 0.035 L (mL)

55 mL = . . . . . . . . . . . . . . . . . . . . . . . . . . . mL 0.035 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

55 mL = . . . . . . . . . mL 0.035 L = . . . . . . . . . L

k 420 L (mL) l 0.99 kL (L)

420 L = . . . . . . . . . . . . . . . . . . . . . . . . . . . mL 0.99 kL = . . . . . . . . . . . . . . . . . . . . . . . . . . . L

420 L = . . . . . . . . . mL 0.99 kL = . . . . . . . . . L

Converting units of timeUnits of time can be converted as follows.1 year = 12 months = 26 fortnights = 52 weeks 1 hour = 60 minutes1 fortnight = 2 weeks 1 minute = 60 seconds

Convert each of the following to the units shown in brackets.a 2 years (months) b 3 fortnights (weeks)

THINK WRITE

a One year = 12 months, so to convert years to months multiply the number of years by 12.

a 2 years = 2 × 12 months= 24 months

b One fortnight = 2 weeks, so to convert fortnights to weeks, multiply the number of fortnights by 2.

b 3 fortnights = 3 × 2 weeks= 6 weeks

3WORKEDExample

136

SkillSHEETanswersTry these3 Convert each of the following to the units shown in brackets.

a 3 years (months) b 2 years (weeks)

3 years = 3 × . . . . . . . . . months 2 years = . . . . . . . . . × . . . . . . . . . weeks

3 years = . . . . . . . . . months 2 years = . . . . . . . . . weeks

c 4 years (fortnights) d 5 fortnights (weeks)

4 years = . . . . . . . . . × . . . . . . . . . fortnights 5 fortnights = . . . . . . . . . × . . . . . . . . . weeks

4 years = . . . . . . . . . fortnights 5 fortnights = . . . . . . . . . weeks

e 48 months (years) f 24 weeks (fortnights)

48 months = 48 ÷ . . . . . . . . . years 24 weeks = . . . . . . . . . ÷ . . . . . . . . . fortnights

48 months = . . . . . . . . . years 24 weeks = . . . . . . . . . fortnights

g 18 months (years) h 5 years (months)

18 months = . . . . . . . . . . . . . . . . . . years 5 years = . . . . . . . . . . . . . . . . . . months

18 months = . . . . . . . . . years 5 years = . . . . . . . . . months

i 4 fortnights (weeks) j 5 hours (minutes)

4 fortnights = . . . . . . . . . . . . . . . . . . weeks 5 hours = . . . . . . . . . . . . . . . . . . minutes

4 fortnights = . . . . . . . . . weeks 5 hours = . . . . . . . . . minutes

k 300 seconds (minutes) l 3 hours (seconds)

300 seconds = . . . . . . . . . . . . . . . . . . minutes 3 hours = 3 × . . . . . . . . . minutes

300 seconds = . . . . . . . . . minutes 3 hours = . . . . . . . . . minutes

300 seconds = ?? minutes 3 hours = . . . . . . . . . × . . . . . . . . . seconds

3 hours = . . . . . . . . . seconds

137

SkillSHEETanswersSkillSHEET 7.9Converting minutes to fractions of an hourTo convert between minutes and hours, remember that:

60 minutes = 1 hour We can change a certain number of minutes to a fraction of an hour by dividing the number of minutes by60.

Try theseWrite each of the following as a fraction of 1 hour.

1 45 minutes = hour 2 30 minutes = hour

45 minutes = hour 30 minutes = hour





Write 20 minutes as a fraction of an hour.

THINK WRITE

Form a fraction by writing 20 as the numerator and 60 as the denominator.

20 minutes = hour

Simplify the fraction by dividing both numerator and denominator by the highest common factor of 20 and 60, that is, divide both by 20.

20 minutes = hour

12060------

213---

WORKEDExample

4560------ .......

60---------

.......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

138

SkillSHEETanswersSkillSHEET 8.1Classifying anglesAngles can be classified according to their size as shown in the table below.

Try theseState the type of each of the following angles according to its size.

1 15° Angle type: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 136° Angle type: . . . . . . . . . . . . . . . . . . . . . . . . . . . .





Size of the angle Name of the angle

Between 0° and 90° An acute angle

90° angle A right angle

Between 90° and 180° An obtuse angle

180° angle A straight line or angle

Between 180° and 360° A reflex angle

360° angle One revolution (or a perigon)

State the type of each of the following angles according to its size.a 72° b 132° c 210°THINK WRITE

a The angle 72° is between 0° and 90°. Find the name for the angle of such size in the table and write it down.

a An acute angle

b Angle 132° is between 90° and 180°. State the name of such an angle.

b An obtuse angle

c Angle 210° is between 180° and 360°, so classify it accordingly.

c A reflex angle

WORKEDExample

139

SkillSHEETanswersSkillSHEET 8.2Classifying triangles according to the lengths of their sidesTriangles can be classified according to their side lengths as follows.

Note that in the diagrams the sides of equal length are shown by identical markings.

Try theseState the name of the triangles shown.

1 2

Triangle name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triangle name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Triangle name Diagram Description

An equilateral triangle All three sides are equal in length.

An isosceles triangle Exactly two sides are equal in length.

A scalene triangle All three sides are of different lengths.

State the name of the triangle shown.

THINK WRITE

Two sides of the triangle have identical markings on them; therefore these two sides are equal in length. State the name of the triangle with exactly two equal sides.

Name: an isosceles triangle

WORKEDExample

140

SkillSHEETanswers3 4


5 6


7 8


9

Triangle name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

SkillSHEETanswersSkillSHEET 8.3Angles in a triangleTriangles can be classified according to their side lengths as follows.

Angle sum of a triangleThe sum of the interior angles in any triangle is 180°.

p + q + r = 180°

Triangle name Diagram Properties

An equilateral triangle All 3 sides are equal in length. All angles are the same and equal to 60°.

An isosceles triangle Two sides are equal in length. The base angles (the angles opposite the equal sides) are equal in size.

A scalene triangle All 3 sides are of different lengths. All 3 angles are of different size.

α β

rp

q

Find the value of the pronumerals in each of the following triangles.a b c

Continued over page

THINK WRITEa The sum of the 3 angles (b, 35° and 58°) must be 180°. Write

this as an equation.a b + 35° + 58° = 180°

Simplify by adding 35° and 58° together. b + 93° − 93° = 180° – 93°Solve for b. b = 87°

b

35º

58ºa

40°

ba

65°

b

1

23

WORKEDExample

142

SkillSHEETanswers

Try these1 Find the value of the pronumeral in each of the following triangles.

a b c

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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d e f

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

THINK WRITEb Form an equation by making the sum of the angles in the

given triangle equal 180°.b a + b + 40° = 180°

The markings on the triangle indicate that it is an isosceles triangle. Therefore, the base angles (angles a and b) are equal in size.

a = b

In the original equation, replace b with a (since they represent the same value) to obtain an equation with one unknown.

a + a + 40° = 180°

Solve for a.Subtract 40° from both sides of the equation.

Divide both sides by 2.

2a + 40° = 180°2a + 40° − 40° = 180° – 40°

2a = 140°a = 140° ÷ 2a = 70°

State the value of b (it is the same as a). b = 70°

c The markings on the sides of the given triangle indicate that it is an isosceles triangle. Therefore, its base angles (angles b and 65°) are equal in size.

c b = 65°

Form an equation by letting the sum of the angles in the triangle equal 180°.

a + b + 65° = 180°

Substitute the value of b into the equation. a + 65° + 65° = 180°Solve for a. a + 130° = 180°

a + 130° − 130° = 180° – 130°a = 50°

1

2

3

4

5

1

2

34

55º

68º x

25º

30º

g

96º

40º

t

60º

60ºk

54º

f

33º 30º

60ºz

143

SkillSHEETanswers2 Find the value of the pronumeral in each of the following right-angled triangles.

a b c

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Find the value of the pronumeral in each of the following triangles.a b c

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Find the value of the pronumeral in each of the following triangles.a b c

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45º

d

b

25º

40ºa

c

55º e

52º64º

n

48º

u k

28º

d

144

SkillSHEETanswersd e f

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32º

t57º

f

70º p

145

SkillSHEETanswersSkillSHEET 8.4Complementary anglesTwo angles are complementary if their sum is 90°. For example, 30° and 60° are complementary, because30° + 60° = 90°. If two angles are complementary, one angle is said to be the complement of the other.

Try these1 Find the complement of each of the following angles.

a 70° b 59° c 36°

x + 70° = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The complement of 70° The complement of 59° The complement of 36°

is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . .

Find the complement of 65°.

THINK WRITE

Let the unknown angle be x. Form an equation by letting the sum of the given angle and its complement be 90°.

x + 65° = 90°

Solve for x by subtracting 65° from both sides of the equation. x + 65° − 65° = 90° – 65°x = 25°

1

2

WORKEDExample 1

Find the value of the pronumeral in the diagram at right.

THINK WRITE

Angles a and 20° are complementary and so add to 90°. State this as an equation.

a + 20° = 90°

Solve for a by subtracting 20° from both sides of the equation.

a + 20° − 20° = 90° – 20°a = 70°

20°a

1

2

WORKEDExample 2

146

SkillSHEETanswersd 12° e 44° f 39°

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The complement of 12° The complement of 44° The complement of 39°

is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Find the value of the pronumeral in each of the following diagrams.

a b c

a + . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60°a

aa

10°x

147

SkillSHEETanswersSkillSHEET 8.5Supplementary anglesTwo angles are supplementary if their sum is 180°. For example, 120° and 60° are supplementary, as120° + 60° = 180°. If the angles are supplementary, one angle is said to be the supplement of the other.

Try these1 Find the supplement of each of the following angles.

a 25° b 47° c 92°

x + 25° = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The supplement of 25° The supplement of 47° The supplement of 92°

is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . .

Find the supplement of 60°.

THINK WRITE

Let the supplement of the given angle be x. The sum of the given angle and its supplement is 180°. State this as an equation.

x + 60° = 180°

Solve for x by subtracting 60° from both sides of the equation. x + 60° − 60° = 180° – 60°x = 120°

1

2

WORKEDExample 1

Find the value of the pronumeral in the diagram at right.

THINK WRITE

Angle b and angle 30° are supplementary and so add up to 180°. State this as an equation.

b + 30° = 180°

To solve for b subtract 30° from both sides of the equation. b + 30° − 30° = 180° – 30°b = 150°

b

30°

1

2

WORKEDExample 2

148

SkillSHEETanswersd 115° e 160° f 39°

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The supplement of 115° The supplement of 160° The supplement of 39°

is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . . is . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Find the value of the pronumeral in each of the following diagrams.

a b c

50° y x80°

155°y

149

SkillSHEETanswersSkillSHEET 8.6Angle sum of a quadrilateralA quadrilateral is a closed 2-dimensional four-sided shape. The sum of the angles in any quadrilateral is 360°.

All quadrilaterals can be subdivided into two groups: parallelograms and other quadrilaterals. Parallelo-grams are quadrilaterals with opposite sides being parallel and include a square, a rectangle, a parallelogramand a rhombus. Other quadrilaterals include a kite, a trapezium and an irregular quadrilateral. The table belowsummarises properties of these quadrilaterals.

Quadrilateral Sides Angles

ParallelogramsSquare Opposite sides are parallel.

All sides are equal in length.All angles are 90°.

Rectangle Opposite sides are parallel and equal in length.

All angles are 90°.

Parallelogram Opposite sides are parallel and equal in length.

Opposite angles are equal in size.

Rhombus Opposite sides are parallel.All sides are equal in length.

Opposite angles are equal in size.

Other quadrilateralsKite Adjacent sides are equal in

length.The angles between the unequal sides are equal in size.

Trapezium One pair of opposite sides are parallel.

150

SkillSHEETanswers

Try these

Find the value of the pronumerals in each of the following quadrilaterals.

1 2

x + 85° + . . . . . . . . . . . . . . +. . . . . . . . . . . . . . = 360° x + . . . . . . . . . . . . . + . . . . . . . . . . . . . + . . . . . . . . . . . . . = . . . . . . . . . . . . .

x + . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Find the value of the pronumeral in each of the following diagrams.a b

THINK WRITEa Form an equation by letting the sum of the angles in the

quadrilateral be 360°.a m + 120° + 110° + 70° = 360°

Solve for m by subtracting 300° from both sides of the equation.

m + 300° = 360°m = 360° – 300° m = 60°

b Since the adjacent sides of the quadrilateral are equal in length, it is a kite. In a kite, angles between unequal sides are equal in size. State this as an equation.

b 2x = 140°

Solve for x by dividing both sides by 2. x = 140° ÷ 2= 70°

The angle sum of a quadrilateral is 360°. State this as an equation.

y + 2x + 30° + 140° = 360°

Replace 2x with 140° as these angles are equal in size. y + 140° + 30° + 140° = 360°Solve for y by subtracting 310° from both sides of the equation.

y + 310° = 360° y = 360° – 310°

= 50°

120°110° 70°

m

140°

30°

2x

y

1

2

1

2

3

45

WORKEDExample

130°

85°

80° x

x105°

xm

135°

151


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7 8

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9

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60°

nm 100° b

a

30°

2x

120°

y

140°

y x

20°

20°

50° x

152

SkillSHEETanswersSkillSHEET 8.7Angle sum of a polygonThe sum of the interior angles of any polygon can be calculated using the formula: 180° × (n – 2) or(180n – 360)°, where n is the number of sides of the polygon.

The sum of the exterior angles in any polygon is 360° (regardless of the number of sides).A regular polygon has all sides equal in length and all angles equal in size. The size of each angle of a

regular polygon can therefore be found by dividing the sum of the interior angles by the number of theangles n, and the size of each exterior angle can be found by dividing 360° by n.

Try these

For each of the following regular polygons find:

i the sum of the interior angles

ii the size of each interior angle

iii the size of each exterior angle.

1 A regular quadrilateral (n = 4) 2 A regular pentagon (n = 5)

i Angle sum = 180° × (n – 2) i Angle sum = 180° × (n – 2)

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

ii One interior angle = . . . . . . . . . . . . . .


= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

iii One exterior angle = . . . . . . . . . . . . . .


= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

For a regular decagon find:a the sum of the interior angles b the size of each interior angle c the size of each exterior angle.THINK WRITEa Write the general formula for the sum of the interior

angles of a polygon.a Angle sum = 180° × (n – 2)

Identify the value of n for a decagon. n = 10Substitute the value of n into the formula and evaluate. Angle sum = 180° × (n – 2)

= 180° × 8 = 1440°

b Since the dodecagon is regular, all interior angles are equal in size. So to find the size of each angle, divide the angle sum by the number of angles.

b One interior angle =

= 144°c The sum of the exterior angles in any polygon is 360°. In a

regular dodecagon there are 10 equal exterior angles. So to find the size of each exterior angle, divide the sum by the number of angles.

c One exterior angle =

= 36°

1

23

144010

------------

36010

---------

WORKEDExample

153

SkillSHEETanswers3 A regular hexagon (n = 6) 4 A regular heptagon (n = 7)


= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .



= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .



= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

5 A regular octagon (n = 8) 6 A regular nonagon (n = 9)


= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . = . . . . . . . . . . . . . . .



= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .



= . . . . . . . . . . . . . . = . . . . . . . . . . . . . .

154

SkillSHEETanswersSkillSHEET 8.8Angles and parallel linesA straight line cutting two or more parallel lines is called a transversal.

When a transversal cuts a set of parallel lines, a number of angles are formed as shown below.

Corresponding anglesAngles that are on the same side of the transversal and are both either above or belowthe parallel lines are called corresponding.

Corresponding angles are equal in size.In the diagram shown, there are four pairs of corresponding angles: a and b, c

and d, e and f, and finally g and h. Angles in each pair are equal in size, that is, a = b, c = d, e = f and g = h.

Corresponding angles are sometimes called ‘F’ angles because of their positionswith respect to the parallel lines and the transversal.

Alternate anglesAngles that are on opposite sides of the transversal, positioned between the parallellines (so that one is below a parallel line while the other is above the second parallel line) are called alternate.

Alternate angles are equal in size.In the diagram shown there are two pairs of alternate angles: a and d, and c and b.

Angles in each pair are equal in size, that is, a = d and c = b.

Alternate angles are also known as ‘Z’ angles.

Co-interior anglesAngles that are on the same side of the transversal, positioned between the parallel lines are called co-interior angles.

Co-interior angles are supplementary (that is, they add to 180º).In the diagram shown at right, there are two pairs of co-interior angles: a and b,

and c and d. Angles in each pair are supplementary; that is, a + b = 180º, and c + d = 180º.

Co-interior angles are often called ‘C’ angles.

ac

bd

eg

f

h

a

b

c

d

a

b

c

d

155

SkillSHEETanswers

Angle properties can be used to test whether the given straight lines are parallel. Straight lines are parallel if:

1. corresponding angles are equal2. alternate angles are equal3. co-interior angles are supplementary.

In the diagrams shown, state whether the given angles are corresponding, alternate or co-interior.a b c

THINK WRITE

a Study the diagram. The shown angles are on the same side of the transversal and are both above the parallel lines (F angles). Name the relationship between the angles.

a Angles x and y are corresponding.

b The given angles are on opposite sides of the transversal and between the parallel lines (2 angles). Name the relationship between the angles.

b Angles x and y are alternate.

c The angles shown are on the same side of the transversal and both are between the parallel lines (C angles). Name the relationship between the angles.

c Angles x and y are co-interior.

x

y

xy

x y

WORKEDExample 1

Find the value of the pronumerals in the following diagram. Give reasons for your answers.

THINK WRITE

Angle x and angle 105° are both to the right of the transversal, positioned between the parallel lines. Therefore, these angles are co-interior and so add up to 180°. State this as an equation.

x + 105° = 180° (as co-interior)

Solve for x by subtracting 105° from both sides of the equation.

x + 105° – 105° = 180° – 105°x = 75°

Angles y and 105° are corresponding, as they are both to the right of the transversal and are both below the parallel lines. Corresponding angles are equal in size.

y = 105° (as corresponding)

Angle z is alternate to 105°, as they are positioned between the parallel lines on opposite sides of the transversal. Alternate angles are equal in size.

z = 105° (as alternate)

xy

z

105°

1

2

3

4

WORKEDExample 2

156

SkillSHEETanswers

Try these

1 In the diagrams shown state whether the given angles are corresponding, alternate or co-interior.

a b c

x and y are . . . . . . . . . . . . . . . . . . . . . . x and y are . . . . . . . . . . . . . . . . . . . . . . x and y are . . . . . . . . . . . . . . . . . . . . . .

d e f


g h i


In each case state whether line AB is parallel to line CD.a b c d

THINK WRITE

a The angles shown are corresponding and are equal in size. State your conclusion.

a The line AB is parallel to CD.

b Alternate angles are equal, so decide whether the two lines are parallel.

b The line AB is parallel to CD.

c The given angles are co-interior. Check whether they are supplementary by adding them together.

c 120° + 60° = 180°

Co-interior angles are supplementary (as they added up to 180°), so state your conclusion.

The line AB is parallel to CD.

d The angles given are in a corresponding position, but are not equal in size. State whether the given lines are parallel.

d The line AB is not parallel to CD.

60°

60°

BA

DC

120°120°

BA

DC

120°60°

BA

DC

60°

55°

BA

DC

1

2

WORKEDExample 3

x

y

x

y

x

y

xy x

y

xy

x

y

x

y x y

157

SkillSHEETanswers2 Find the value of the pronumerals in the following diagrams. Give reasons for your answers

a b c

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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d e f

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g h i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 In each case state whether line AB is or is not parallel to line CD.

a b c

The line AB is The line AB . . . . . . . . . . . . . . The line AB . . . . . . . . . . . . . .

. . . . . . . . . . . . . . to CD. parallel to CD. parallel to CD.

ts

100°

x

140°x y

110°

xy

36°x

zy50°

5y95°

x 7y

70° x + 10°30°

3x

30°

65°

65°

BA

DC 100° 70°

B

A

D

C

60°55° B

A

D

C

158

SkillSHEETanswersd e f

The line AB . . . . . . . . . . . . . . The line AB . . . . . . . . . . . . . . The line AB . . . . . . . . . . . . . .

parallel to CD. parallel to CD. parallel to CD.

g h i

The line AB . . . . . . . . . . . . . . The line AB . . . . . . . . . . . . . . The line AB . . . . . . . . . . . . . .

parallel to CD. parallel to CD. parallel to CD.

70°70°

B

A

D

C

110°

80°

B

A

DC

85°85°

A

B

C

D

50°

50° BA

DC120°

60°BA

DC100°110°

B

A

D

C

159

SkillSHEETanswersSkillSHEET 8.9Angle relationshipsWhen two lines cross each other, two pairs of vertically opposite angles are formed. Vertically oppositeangles are equal in size. In the figure shown below ∠AOB = ∠DOC and ∠AOD = ∠BOC.

Angles on a straight line are called supplementary. Supplementary angles add up to 180°. In the figureshown below ∠AOB + ∠BOC = 180°.

O

D

A

C

B

AO

B

C

For each of the following, name the relationship between the angles shown and write this relationshipas an algebraic statement. (Do not simplify.)a b

THINK WRITE

a The angles shown are formed by two intersecting lines and are opposite each other. State the angle relationship.

a The given angles are vertically opposite.

Vertically opposite angles are equal. Write this as an algebraic statement.

x + 120 = 2x − 40

b The angles shown are on a straight line. Name the corresponding relationship.

b The given angles are supplementary.

Supplementary angles add up to 180°. Write this as an algebraic statement.

x + 7x − 10 = 180

x + 120

2x – 40

7x – 10x

1

2

1

2

WORKEDExample

160

SkillSHEETanswersTry theseFor each of the following name the relationship between the angles shown and write this relationship as analgebraic statement. (Do not simplify.)

1 2 3

4 5

6 7 8

9 10

2x

4x – 200

3x x + 20 x

3x – 20

2x7x

2x + 10

5x – 250

x + 302x – 55

3x + 10

2x + 40

3x + 4 xx

7x – 50

2x + 80

x

x

2x – 5

x + 10

2x

161

SkillSHEETanswersSkillSHEET 8.10More angle relationships

Angles on a straight line add to 180°.

Angles at a point add to 360°.

When two lines intersect, two pairs of vertically opposite angles are formed. Vertically opposite angles are equal in size.

In the diagram shown at right, a = b and c = d, as they are vertically opposite.

ca

a + b + c = 180°

b

a + b + c + d + e + f = 360°

abc

d ef

a

c d

b

Find the value of the pronumeral in each of the following diagrams.

a b c

THINK WRITE

a Angle b and angle 30° are supplementary and so add up to 180°. State this as an equation.

a b + 30° = 180°

To solve for b subtract 30° from both sides of the equation.

b + 30° − 30° = 180° – 30°b = 150°

Angle c and angle 30° are vertically opposite and therefore are equal in size.

c = 30°

b Angles a, 40° and 70° are on a straight line and so add up to 180°. State this as an equation.

b a + 40° + 70° = 180°

To solve for a, first add 40° and 70° together. a + 110° = 180°Subtract 110° from both sides of the equation. a + 110° − 110° = 180° – 110°

a = 70°c Angles 35°, 120°, 35°, 40° and c are at a point

and so add up to 360°. State this as an equation.

c 35° + 120° + 35° + 40° + c = 360°

Simplify by adding the numbers on the left side of the equation together.

230° + c = 360°

Subtract 230° from both sides of the equation to find the value of c.

230° − 230° + c = 360° – 230°230° − 230° + c = 130°

b

c

30°

a40° 70° 35°

35°

120°40°

c

1

2

3

1

23

1

2

3

WORKEDExample

162

SkillSHEETanswersTry these

Find the value of the pronumeral in each of the following diagrams.

1 2 3

4 5 6

80°a

120°x

y

80°118°

62° x

y

80°120°3x

xx

xx

3x

50°

60°3m

163

SkillSHEETanswersSkillSHEET 8.11Measuring and drawing linesMeasuring the length of a lineWhen measuring the length of sides of different shapes with a ruler, it is important to remember that thelarge marks (with numbers next to them) represent centimetres (cm) and the small marks represent milli-metres (mm). Each centimetre contains 10 millimetres.

To measure the length of a line (or the side of a shape) in millimetres, follow these steps.1. Position the ruler along the line so that the zero mark is at one end of it.2. Look at the other end of the line. Note the number of centimetres, indicated by the nearest large mark to

the left of the end-point of the line. 3. Count the number of millimetres (that is, the small marks) from the last large mark to the end of the line.4. Multiply the number of centimetres by 10, and then add the number of millimetres.

Try these1 Find the lengths of the following lines in millimetres.

a b

Find the length of the following line in millimetres.

THINK WRITE

Position the ruler along the line so that the zero mark is at one end of it.

The nearest centimetre mark to the end of the line shows 3 cm. From the 3-cm mark, there are 4 millimetre marks to the end of the line.

AB = 3 cm 4 mm

Convert 3 centimetres to millimetres by multiplying by 10 and then add 4 mm to find the total length of the line.

= 3 × 10 + 4 = 30 + 4 = 34 mm

A B

1 A

0 cm 1 2 3 4

B

2

3

1WORKEDExample

A B

Length of the line = . . . . . . . . . . . . . .

A

B


164

SkillSHEETanswersc d

e f

g h

i

Ruling a line to a required lengthTo draw a line of a required length we use a ruler. On the ruler, large marks with numbers next to themrepresent centimetres, while small marks represent millimetres.

A

B


A

B


A B

Length of the line = . . . . . . . . . . . . . . A BLength of the line = . . . . . . . . . . . . . .

A

B


A

B


A

B


Centimetremarks

Millimetremarks

10 2 3 4 5 6 7 8 9

165

SkillSHEETanswers

Try these2 Rule lines of the following lengths:

a 4 cm b 7 cm c 12 cm d 48 mm e 72 mm

f 80 mm g 66 mm h 9.2 cm i 3.5 cm j 5.8 cm

Rule a line with a length of a 6 cm b 54 mm

THINK DRAW

a Place your ruler on the page where you want to draw a line. Draw a line (by moving your pencil along the ruler) from the large mark with 0 next to it to the large mark with 6 next to it. The resultant line is 6 cm long.

a

b 54 mm = 5 cm and 4 mm. First draw a line from the large mark with 0 next to it to the large mark with 5 next to it. (This gives 5 cm of length.) Now extend the line to the fourth small mark. (This gives another 4 mm of length.) The resultant line is 54 mm long.

b

10 2 3 4 5 6 7 8 9

10 2 3 4 5 6 7 8 9

2WORKEDExample

166

SkillSHEETanswersSkillSHEET 8.12Constructing angles with a protractorThe following worked example shows the procedure of constructing an angle using a protractor.

Try theseConstruct each of the following angles using a protractor.

1 90° 2 60°

3 72° 4 45°

Construct an angle of 75°, using a protractor.

THINK DRAW

Draw a baseline. Position your protractor so that its centre is at one end point of the baseline and its 0° mark is at the other.

Locate a 75° angle (use the scale with the 0 on the baseline) and mark it with a small dot.

Remove the protractor. Join the vertex of the angle (that is, the end point of the baseline, where the centre of the protractor was) with the 75° mark using a straight line.

1

0°

2

0°

75°

3

WORKEDExample

167

SkillSHEETanswers5 30° 6 120°

7 150° 8 36°

9 40° 10 24°

168

SkillSHEETanswersSkillSHEET 8.13Using a pair of compasses to draw circlesThe most basic use of a pair of compasses is to draw a perfect circle.

Your compasses consist of a point and a place to insert a pencil. The point of the compasses is placedwhere the centre of the circle is to be. The circle is then drawn by moving the pencil around this centre. Thedistance between the point of the compasses and the end of the pencil is equal to the radius of the circle.

Construct a circle with a radius of 3 cm.

THINK DRAW

Set the legs of the compasses a distance of 3 cm apart.

Place the point of the compasses where the centre of the circle is to be, and rotate the pencil around the centre to draw the circle.

1

cm 1 2 3 4

2

WORKEDExample

169

SkillSHEETanswersTry theseIn the space below, draw a circle with a radius of:

1 5 cm 2 2 cm

3 24 mm 4 37 mm

170

SkillSHEETanswersSkillSHEET 9.1Rounding to one decimal placeTo round a decimal fraction to 1 decimal place, follow these steps.1. Consider the digit in the second decimal place. 2. If it is less than 5, simply omit this digit and all digits that follow.3. If it is 5 or larger, add 1 to the preceding digit (that is, to the digit in the first decimal place) and omit all

digits that follow.

Note that the sign ≈ is read as ‘is approximately equal to’.

Try these

Round each of the following decimal fractions to 1 decimal place.

1 4.47 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.77 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 1.209 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.569 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 0.913 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7.235 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 0.5502 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10.2978 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 6.3004 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 0.394 12 ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Round each of the following numbers to 1 decimal place.a 5.371 b 8.7234

THINK WRITE

a Consider the digit in the second decimal place. It is 7, which is greater than 5. So add 1 to the preceding digit (3) and omit all digits that follow.

a 5.371 ≈ 5.4

b The digit in the second decimal place is 2, which is less than 5. So simply omit this digit and all digits that follow (that is, omit 2, 3 and 9).

b 8.7239 ≈ 8.7

1WORKEDExample

171

SkillSHEETanswersSkillSHEET 9.2Measuring angles with a protractorAngles smaller than or equal to 180° can be measured with a semicircular protractor, as shown in thefollowing worked example.

A circular protractor could be used in exactly the same manner as the semicircular one to measure the anglein the above worked example.

A circular protractor is also used in the same way to measure a reflex angle (that is, an angle between180° and 360°). However, if a circular protractor is unavailable, a reflex angle can be measured using asemicircular protractor. To do this, we measure the smaller angle first and then subtract it from 360°.

Find the size of the following angle.

THINK WRITE

Position your protractor so that its centre is at the vertex and its baseline (with the 0) coincides with one of the arms of the angle.

Read the size of the angle, indicated by the second arm. (Make sure you use the scale that begins with 0 where the first arm is.) Write your answer.

The size of the given angle is 60°.

1

60°

0°

2

1WORKEDExample

60°

0°

172

SkillSHEETanswers

Try these1 Find the size of each of the following angles.

a b

c d

e

Find the size of the following angle, using a semicircular protractor.

THINK WRITE

First measure the obtuse angle. Position the protractor so that its centre is at the vertex, its 0 is at one arm of the angle and the angle size increases towards the other arm of the obtuse angle.

Read the size of the obtuse angle, as indicated by the second arm of the angle. Write down the answer.

The obtuse angle = 100°

To find the size of the reflex angle, subtract the obtuse angle from 360°.

The reflex angle = 360° − 100°The reflex angle = 260°

1

100°

0°

2

3

2WORKEDExample

The required angle = . . . . . . . . . . . . . .





173

SkillSHEETanswers2 Find the size of each of the following angles, using a semicircular protractor.

a b

c d

e

The obtuse angle = . . . . . . . . . . . . . .

The reflex angle = 360° − . . . . . . . . . . . . . .

The reflex angle = . . . . . . . . . . . . . .

The obtuse angle = . . . . . . . . . . . . . .

The reflex angle = 360° − . . . . . . . . . . . . . .

The reflex angle = . . . . . . . . . . . . . .

The acute angle = . . . . . . . . . . . . . .

The reflex angle = 360° − . . . . . . . . . . . . . .

The reflex angle = . . . . . . . . . . . . . .

The acute angle = . . . . . . . . . . . . . .

The reflex angle = 360° − . . . . . . . . . . . . . .

The reflex angle = . . . . . . . . . . . . . .

The acute angle = . . . . . . . . . . . . . .

The reflex angle = 360° − . . . . . . . . . . . . . .

The reflex angle = . . . . . . . . . . . . . .

174

SkillSHEETanswersSkillSHEET 9.3Measuring the length of a lineWhen measuring the length of sides of different shapes with a ruler, it is important to remember that thelarge marks (with numbers next to them) represent centimetres (cm) and the small marks represent milli-metres (mm). Each centimetre contains 10 millimetres.

To measure the length of a line (or the side of a shape) in millimetres, follow these steps.Step 1 Position the ruler along the line so that the zero mark is at one end of it.Step 2 Look at the other end of the line. Note the number of centimetres, indicated by the nearest large

mark to the left of the end-point of the line. Step 3 Count the number of millimetres (that is, the small marks) from the last large mark to the end of

the line.Step 4 Multiply the number of centimetres by 10, and then add the number of millimetres.

Find the lengths of the following lines in millimetres.

Try these

1 2

Find the length of the following line in millimetres.

THINK WRITE

Position the ruler along the line so that the zero mark is at one end of it.

The nearest centimetre mark to the end of the line shows 3 cm. From the 3-cm mark, there are 4 millimetre marks to the end of the line.

AB = 3 cm 4 mm

Convert 3 centimetres to millimetres by multiplying by 10 and then add 4 mm to find the total length of the line.

= 3 × 10 + 4 = 30 + 4 = 34 mm

A B

1 A

0 cm 1 2 3 4

B

2

3

WORKEDExample

A B


A

B


175


5 6

7 8

9

A

B


A

B


A B

Length of the line = . . . . . . . . . . . . . . A BLength of the line = . . . . . . . . . . . . . .

A

B


A

B


A

B


176

SkillSHEETanswersSkillSHEET 9.4Multiplying and dividing by powers of 10To multiply a whole number by a power of 10, add as many zeros to that number as there are in the powerof 10. For example, to multiply by 10, add one zero and to multiply by 100, add two zeros.

To multiply a decimal fraction by powers of 10, move the decimal point one place to the right for eachzero in the power of 10. For example, to multiply by 10, move the decimal point one place to the right,while to multiply by 1000 move it three places to the right. Note that if there are not enough digits after thedecimal point, we can always add extra zeros.

To divide a whole number or a decimal fraction by powers of 10, move the decimal point to the left oneplace for each zero in the power of 10. Note that although a whole number does not have a decimal point,we can always add it at the end of the number (for example, 35 and 35. are the same number). Also notethat, if there are not enough digits to move the decimal point the required number of places, we can alwaysadd extra zeros.

Calculate each of the following.a 72 ¥ 10 b 540 ¥ 1000 c 5.67 ¥ 10 d 0.7 ¥ 100

THINK WRITE

a To multiply a whole number by 10, add one 0 to the number, as there is only one zero in 10.

a 72 × 10 = 720

b To multiply a whole number by 1000, add three zeros, since there are 3 zeros in 1000.

b 540 × 1000 = 540 000

c To multiply a decimal by 10, move the decimal point one place to the right (as there is one zero in 10).

c 5.67 × 10 = 56.7

d To multiply a decimal by 100, we need tomove the decimal point two places to the right.However, as there is only one digit after thedecimal, we add a zero first (to create twodecimal places) and then move the decimalpoint. Note that we write the answer as 70,rather than 070.

d 0.7 × 100

= 0.70 × 100= 70

1WORKEDExample

177

SkillSHEETanswers

Try these


a 23 × 10 b 45 × 10 c 235 × 10 d 530 × 10

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

e 2500 × 10 f 71 × 100 g 531 × 100 h 300 × 100

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

i 25 × 1000 j 710 × 1000 k 2 × 10 000 l 70 × 10 000

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .


a 2.56 × 10 b 7.6 × 10 c 0.98 × 10 d 3.49 × 100

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

Calculate each of the following.a 234 ∏ 100 b 350 ∏ 1000 c 75.6 ∏ 10 d 4.1 ∏ 1000

THINK WRITE

a Put a decimal point at the end of the whole number.

a 234 ÷ 100 = 234. ÷ 100

To divide by 100, move the decimal point 2 places to the left, as there are 2 zeros in 100.

234 ÷ 100 = 2.34

b Put a decimal point at the end of the whole number.

b 350 ÷ 1000 = 350. ÷ 1000

There are 3 zeros in 1000, so to divide by 1000 move the decimal point 3 places to the left. You may omit the zero at the end of the resulting decimal (as 0.350 = 0.35).

350 ÷ 1000 = 0.350350 ÷ 1000 = 0.35

c To divide by 10, move the decimal point one place to the left (as there is one zero in a ten).

c 75.6 ÷ 10 = 7.56

d To divide by 1000, move the decimal point three places to the left, since there are three zeros in 1000. (Add two extra zeros in front of the number as you go.)

d 4.1 ÷ 1000 = 0.0041

1

2

1

2

2WORKEDExample

178

SkillSHEETanswerse 2.6 × 100 f 70.1 × 100 g 0.2 × 100 h 5.321 × 1000

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

i 10.2 × 1000 j 0.758 × 1000 k 2.5 × 10 000 l 3.576 × 10 000

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .


a 128 ÷ 10 b 7560 ÷ 10 c 3400 ÷ 10 d 2050 ÷ 10

= 128. ÷ 100 = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . .

e 15 ÷ 100 f 7 ÷ 100 g 560 ÷ 100 h 7210 ÷ 100

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

i 3 ÷ 1000 j 75 ÷ 1000 k 600 ÷ 1000 l 2500 ÷ 10 000

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .


a 9.2 ÷ 10 b 52.3 ÷ 10 c 0.5 ÷ 10 d 8.19 ÷ 100

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

e 4.9 ÷ 100 f 123.4 ÷ 100 g 0.3 ÷ 100 h 71.1 ÷ 1000

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

i 155.6 ÷ 1000 j 4.25 ÷ 1000 k 75.3 ÷ 10 000 l 1000.5 ÷ 10 000

= . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . .

179

SkillSHEETanswersSkillSHEET 9.5Converting units of lengthTo convert units of length, the following chart may be useful.

Conversion of units of length

Try theseConvert each of the following to the units given in brackets.

1 70 m [cm] 2 350 m [cm] 3 5 m [cm]

70 × . . . . . . . = . . . . . . . 350 × . . . . . . . = . . . . . . . . . . . . . . = . . . . . . .

70 m = . . . . . . . cm 350 m = . . . . . . . cm 5 m = . . . . . . . cm

4 10 km [cm] 5 400 km [cm] 6 5000 km [m]

10 × . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 km = . . . . . . . cm .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

mkm mmcm

× 1000 × 100 × 10

÷ 10÷ 100÷ 1000

Convert each of the following to the units given in brackets.a 20 m [cm] b 3000 m [km]

THINK WRITE


a 20 × 100 = 2000



b 3000 ÷ 1000 = 3


1

2

1

2

WORKEDExample

180

SkillSHEETanswers7 200 cm [m] 8 500 mm [cm] 9 8000 m [km]

200 ÷ . . . . . . . = . . . . . . . 500 ÷ . . . . . . . = . . . . . . . . . . . . . . ÷ . . . . . . . = . . . . . . .

200 cm = . . . . . . . m 500 mm = . . . . . . . cm 8000 m = . . . . . . . km

10 200 000 cm [km] 11 50 cm [m] 12 750 m [km]

200 000 ÷ . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200 000 cm = . . . . . . . km .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 78 m [cm] 14 25 km [cm] 15 8 m [cm]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

SkillSHEETanswersSkillSHEET 9.6Substitution into a formulaTo substitute the values of pronumerals into a formula, replace the pronumerals with their correspondingvalues. When all but one pronumeral in the formula are replaced with numbers, the value of the remainingpronumeral can be evaluated. Order of operations must be observed at all times while evaluating.

Try theseSubstitute the given values of the pronumerals into each formula and hence find the value of the unknownpronumeral. Note: All measurements are in centimetres.

1 P = 2(l + w), l = 3, w = 7 2 P = 2(l + w), l = 2.5, w = 1.4

P = 2 × (. . . . . . . + . . . . . . . ) P = 2 × (2.5 + . . . . . . . )

P = 2 × . . . . . . . P = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P = . . . . . . . . . . . . . . . . . . . . . . . . . . . . P = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 A = l × w, l = 12, w = 5 4 A = l2, l = 1.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 C = 2πr, π = 3.14, r = 8 6 C = πD, π = , D = 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 A = πr2, π = 3.14, r = 9 8 A = bh, b = 9, h = 5.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 A = bh, b = 8.6, h = 2.45 10 A = (a + b), a = 2.8, b = 5, h = 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Substitute the given values of the pronumerals into the formula and hence find the value of P.P = 2(l + w), l = 4 cm, w = 8 cm

THINK WRITE

Write the question. P = 2(l + w), l = 4, w = 8, P = ?Replace l with 4 and w with 8. P = 2(4 + 8)To find the value of P, perform addition inside the brackets first and then multiply the result by 2.

P = 2 × 12P = 24 cm

123

WORKEDExample

227------

12---

h2---

182

SkillSHEETanswersSkillSHEET 9.7Solving equationsTo solve an equation, follow the steps outlined below.Step 1 Consider the operations (and their order) that were performed in the process of forming the

equation.Step 2 Perform inverse operations on both sides of the equation in reverse order, until the pronumeral is

left by itself on one side of the equation.

For example, if in the process of forming an equation the unknown number is first multiplied by 3 and then1 is added, to solve the equation we need to first subtract 1 from both sides of the equation and then divideboth sides by 3. That is, if the order of operations is × 3, + 1, then the operations in reverse order must be− 1, ÷ 3.

Solve each of the following equations.a 2x + 5 = 13 b 5a – 4 = –24

THINK WRITE

a Write the equation. a 2x + 5 = 13In the process of forming this equation, the unknown number x is first multiplied by 2 and then 5 is added. To solve the equation we need to perform inverse operations in reverse order; that is, subtract 5 and then divide by 2. So subtract 5 from both sides of the equation.

2x + 5 – 5 = 13 – 52x = 8

Divide both sides of the equation by 2 to find the value of x. =

x = 4

b Write the equation. b 5a – 4 = –24The unknown number a is first multiplied by 5 and then 4 is subtracted. To solve this equation we need to perform inverse operations in reverse order; that is, add 4 and then divide by 5. So add 4 to both sides of the equation.

5a – 4 + 4 = –24 + 45a = –20

Divide both sides of the equation by 5 to find the value of x. Left sid =

a = –4

12

3 2x2

------ 82---

12

3 5a5

------ 20–5

---------

WORKEDExample

183

SkillSHEETanswersTry theseSolve each of the following equations.

1 2x + 9 = 11 2 3x – 5 = 13

2x + . . . . . . . – 9 = 11 – . . . . . . . 3x – . . . . . . . + . . . . . . . = 13 + 5

2x = . . . . . . . . . . . . . . = 18

= =

x = . . . . . . . x = . . . . . . .

3 5a + 2 = 17 4 7m – 4 = 31

. . . . . . . + . . . . . . . – 2 = . . . . . . . – . . . . . . . . . . . . . . – . . . . . . . + . . . . . . . = . . . . . . . + . . . . . . .

. . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 9c + 18 = 0 6 4y – 7 = 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 6p + 5 = –13 8 10d – 22 = –2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 8k + 15 = –17

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2x2

------ .......

2-------- .......

.......-------- 18

.......--------

.......

.......-------- .......

.......--------

184

SkillSHEETanswersSkillSHEET 9.8Area of squares, rectangles and trianglesThe area of many plane figures can be found by using a formula. The table below shows the formula for thearea of some common shapes.

Shape Formula

1. Square A = l2, where l is a side length.

2. Rectangle A = l × w, where l is the length and w is the width.

3. Triangle A = bh, where b is the base length and h the height.

l

l

w

b

h

12---

Find the area of a rectangle with dimensions shown below.

THINK WRITE

Write the formula for the area of a rectangle. A = l × w

Identify the values of l and w. l = 4 and w = 3

Substitute the values of l and w into the formula and evaluate. Include the appropriate units.

A = 4 × 3= 12 cm2

4 cm

3 cm

1

2

3

1WORKEDExample

185

SkillSHEETanswers

Try these1 Find the area of each of the squares below.

a b c

A = l2 A = l2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= 82 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

d e f

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Find the area of the triangle at right.

THINK WRITE

Write the formula for the area of a triangle. A = bh

Identify the values of b and h. b = 10, h = 4.5

Substitute the values of b and h into the formula.

A = × 10 × 4.5

Evaluate. (Since one of the values is even, halve it first if you are not using a calculator, to make calculations easier.) Remember to include the correct unit (cm2).

= 5 × 4.5= 22.5 cm2

10 cm

4.5 cm

112---

2

3 12---

4

2WORKEDExample

8 cm29 mm

3.6 km

2.9 m

3.7 cm 12.5 cm

186

SkillSHEETanswers2 Find the area of each of the rectangles below.

a b c

A = l × w A = l × w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= 9 × 3 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

d e f

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Find the area of each of the triangles below.

a b c

A = bh A = bh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= × 9 × 12 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

d e f

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 m

9 m27 mm

38 mm

47 cm

62 cm

49.7 km2.2 km 3.85 m

34 m

6.3 m

6.4 m

12 m

9 m

6.2 cm

9.4 cm 76 mm

82 mm

12--- 1

2---

12---

4.2 m

9.7 m

8.4 km

6.3 km3.7 m

187

SkillSHEETanswersSkillSHEET 9.9Substitution into area formulasSubstituting the values of pronumerals into a formula means replacing the pronumerals with their corre-sponding values. When all but one of the pronumerals in the formula are replaced with numbers, the valueof the remaining pronumeral can be evaluated. Order of operations must be observed at all times while eval-uating.

Try theseSubstitute the values of the pronumerals into each of the following formulas and hence find the value of A.Note: All measurements are given in cm.

1 A = l2, l = 7 2 A = l2, l = 1.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 A = l × w, l = 12, w = 5 4 A = l × w, l = 6.4, w = 7.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 A = πr2, π = 3.14, r = 9 6 A = πr2, π = , r = 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Substitute the values of the pronumerals into the formula, and hence find the value of A.A = (a + b)h, a = 4 cm, b = 8 cm, h = 5 cm

THINK WRITE

Write the given information. A = (a + b)h, a = 4, b = 8, h = 5, A = ?

Replace a with 4, b with 8, and h with 5. A = × (4 + 8) × 5

To find the value of A, add 4 and 8 first, then divide the sum by 2 and finally multiply the result by 5.

A = × 12 × 5

A = 6 × 5A = 30 cm2

12---

112---

212---

312---

WORKEDExample

227------

188

SkillSHEETanswers7 A = b × h, b = 8.6, h = 2.45 8 A = b × h, b = 9, h = 5.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 A = (a + b)h, a = 6, b = 10, h = 7 10 A = (a + b)h, a = 2.8, b = 5, h = 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12---

12--- 1

2---

189

SkillSHEETanswersSkillSHEET 9.10Total surface area of cubes and rectangular prismsThe total surface area (TSA) of a cube can be found using the formula: TSA = 6l2, where l is the length ofthe edge of the cube.

The total surface area (TSA) of a rectangular prism is given by: TSA = 2(lh + lw + wh), where l is thelength, w is the width and h is the height of the prism.

Find the total surface area of each of the solids shown.a b

THINK WRITE

a The solid shown is a cube. Write down the formula for the total surface area of the cube.

a TSAcube = 6l2

State the value of l (the length of the edge of the cube).

l = 5

Substitute 5 for l into the formula and evaluate.

TSA = 6 × (5)2

TSA = 6 × 25 TSA = 150 cm2

b The solid shown is a rectangular prism. Write the appropriate TSA formula.

b TSArectangular prism = 2(lh + lw + wh)

Identify the values of the pronumerals. l = 10, w = 8 , h = 5Substitute the values of the pronumerals into the formula and evaluate.

TSA = 2(10 × 5 + 10 × 8 + 8 × 5)TSA = 2(50 + 80 + 40)TSA = 340 cm2

5 cm 10 cm8 cm

5 cm

1

2

3

1

23

WORKEDExample

190

SkillSHEETanswersTry theseFind the total surface area of each of the solids shown.

1 2

TSAcube = ....... TSArectangular prism = .......

l = ....... l = ......., w = ......., h = .......

TSA = 6 × (.......)2 TSA = 2 (....... × ....... + ....... × ....... + ....... × .......)

STA = ....... × ....... STA = 2 (....... + ....... + .......)

STA = ....... cm2 SAT = ....... cm2

3 4

TSA ....... = ....... ..........................................

............................ ..........................................

TSA = .............. ..........................................

STA = .............. ..........................................

STA = .............. ..........................................

5 6

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

12 cm

2 cm

1 cm

8 cm

7.5

m

2 cm

2 cm

12.4 cm

1.3 mm

50 cm

33 cm

40 cm

191


.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

8 m

2.1 m

8 m

8.4 mm

192

SkillSHEETanswersSkillSHEET 9.11Total surface area of triangular prismsAny triangular prism has two identical triangular faces and three rectangular faces (which may, or may notbe the same). To find the total surface area (TSA) of a triangular prism, find the area of each individual faceand add them all together.

The area of a triangular face can be found using the formula A = × b × h, where b is the base lengthand h is the height of the triangle.

The area of a rectangular face can be found using the formula A = l × w, where l is the length and w isthe width of the rectangle.

12---

Find the total surface area of each of the following prisms.a b

THINK WRITE

a The front and back faces of the given prism are identical triangles. Write the formula for finding the area of a triangle.

a A = × b × h

Identify the values of the pronumerals. b = 6, h = 5.2Substitute the values of the pronumerals into the formula and evaluate.

Afront = Aback = × 6 × 5.2

Afront = Aback = 15.6 mm2 The other three faces are identical rectangles (because the front face is an equilateral triangle). Write the formula for the area of a rectangle.

A = l × w

Identify the values of the pronumerals. l = 10, w = 6Substitute the values of the pronumerals into the formula and evaluate.

Aleft side = Aright side = Abottom = 10 × 6

Aleft side = Aright side = Abottom = 60 mm2 To find the total surface area, add the areas of all faces together. Since there are two identical triangular faces and three identical rectangular faces, a shortcut can be used. That is, multiply the area of the triangle by 2 and the area of the rectangle by 3, and then add.

TSA = 2 × 15.6 + 3 × 60STA = 31.2 + 180STA = 211.2 mm2

Continued over page

112---

23 1

2---

4

56

7

WORKEDExample

12 cm

20 cm5 cm

13 cm

5.2 mm

6 mm

10 mm

193

SkillSHEETanswers

THINK WRITE

b The front and back faces of the given prism are identical triangles. Write the formula for finding the area of a triangle.

b A = × b × h

Identify the values of the pronumerals. (Since the triangle is right-angled, the height is given by one of the shorter sides.)

b = 5, h = 12

Substitute the values of the pronumerals into the formula and evaluate.

Afront = Aback = × 5 × 12

Afront = Aback = 30 cm2

The other three faces are rectangles. Write the formula for the area of a rectangle.

A = l × w

In this prism all rectangular faces are different, so we need to find the area of each face separately. Identify the values of the pronumerals for each face.

Left side: l = 20, w = 13Right side: l = 20, w = 12Bottom: l = 20, w = 5

For each face substitute the values of the pronumerals into the formula and evaluate the area.

Aleft side = 20 × 13 = 260 cm2

Aright side = 20 × 12 = 240 cm2

Abottom = 20 × 5 = 100 cm2

To find the total surface area, add the areas of all faces together. (As there are two identical triangular faces, simply multiply the area of the triangle by 2.)

TSA = 2 × 30 + 260 + 240 + 100STA = 660 cm2

112---

2

3 12---

4

5

6

7

194

SkillSHEETanswersTry theseFind the total surface area of each of the following prisms.

1 2

Afront = Aback = × b × h Afront = Aback = .......

b = ......., h = ....... ..........................................

Afront = Aback = × ....... × ....... Afront = Aback = .......

Afront = Aback = .............. Afront = Aback = ..............

Aleft side = Aright side = Abottom = l × w Aleft side = Aright side = Abottom = ..............

l = ......., w = ....... ..........................................

Aleft side = Aright side = Abottom = ....... × ....... Aleft side = Aright side = Abottom = ..............

Aleft side = Aright side = Abottom = .............. Aleft side = Aright side = Abottom = ..............

TSA = 2 × ....... + 3 × ....... TSA = ............................

STA = ....... + ....... TSA = ..............

STA = .............. TSA = ..............

10 mm

20 mm

8.7 mm

3.5 cm

4 cm

7 cm

12---

12---

195


..........................................

..........................................

..........................................

..........................................

..........................................

..........................................

..........................................

..........................................

..........................................

..........................................

..........................................

5 6

Afront = Aback = ....... ..........................................

.......................................... ..........................................

Afront = Aback = ....... ..........................................

Afront = Aback = .............. ..........................................

Afront = Aback = .............. ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

Aleft side = .............. ..........................................

Aright side = .............. ..........................................

Abottom = .............. ..........................................

TSA = ............................ ..........................................

STA = ............................ ..........................................

8 cm

12 cm6.9 cm

Afront = Aback = × w × h

w = ......., h = .......

Afront = Aback = × ....... × .......

Afront = Aback = ..............

A = l × w

Left side: l = ......., w = .......

Right side: l = ......., w = .......

Bottom: l = ......., w = .......

Aleft side = .............. × .............. = ..............

Aright side = .............. × .............. = ..............

Abottom = .............. × ............. = ..............

TSA = 2 × ....... + ....... + ....... + .......

STA = ............................

12---

12---

3 cm

4 cm

5 cm

8 cm

25 mm

15 cm

8 mm

20 mm

41 cm

40 cm9 cm44 cm

196

SkillSHEETanswersSkillSHEET 9.12Volume of cylindersThe formula V = AH can be used for any solid that has a uniform cross-sectional area.

For a cylinder, the cross-section is a circle whose area is given by A = πr2, where r is the radius of thecircle. Therefore, the formula for the volume of a cylinder is V = πr2H, where r is the radius of the cylinderand H is its height. (Use the value for π from your calculator.)

Note that the actual height of the cylinder does not always represent the true meaning of the word; thatis, height is not necessarily vertical. (In fact it is only vertical if the cylinder stands on one of its circularfaces.) Height should be thought of as the dimension which is perpendicular to the cross-section of theobject.

Find the volume of the cylinder shown, correct to 2 decimal places.

THINK WRITE

Write the formula for the volume of a cylinder.

V = πr2H

Identify the values of the pronumerals. Note that we are given the diameter rather than the radius. In order to obtain the radius, divide the diameter by 2 (since d = r × 2).

r = = 5, H = 12

Substitute the values of the pronumerals into the formula and evaluate. State your answer correct to 2 decimal places, and include the appropriate units.

V = π × 52 × 12V = 942.48 cm3

12 cm

10 cm

1

2 102

------

3

1WORKEDExample

197

SkillSHEETanswersTry theseFind the volume of the cylinders shown, correct to 2 decimal places.

1 2

V = πr2H V = ...................................

r = ......., H = ....... V = ...................................

V = π × ....... × ....... V = ...................................

V = ................................... V = ...................................

3 4

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

5 6

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

14 m

4 m

11 cm

20 cm

15 cm

5 cm 16 mm

60 mm

9 m

8 m

17 cm

21 cm

198

SkillSHEETanswersSkillSHEET 9.13Volume of triangular prismsThe volume (V) of any prism can be calculated using the formula V = AH, where A is the area of the cross-section and H is the height of a prism.

In a triangular prism, the cross-section is a triangle whose area is given by A = × b × h, where b is the

base length and h is the height of the triangle. Therefore, the formula for the volume of a triangular prism

can be written as V = × b × h × H, where b and h are the base length and the height of the triangular face

respectively, and H is the height of the prism.Note that the actual height of the prism does not always represent the true meaning of the word; that is,

height is not necessarily vertical. (In fact it is only vertical if the prism stands on one of its triangular faces.)Height should be thought of as the dimension which is perpendicular to the cross-section of the prism.

12---

12---

Find the volume of the triangular prism shown.

THINK WRITE

Write the formula for the volume of a triangular prism.

V = × b × h × H

Identify the values of the pronumerals. Remember that h is the height of the triangular face and H is the height of the prism. Since the given prism does not stand on its triangular face, its height (H) is not vertical; it is the dimension perpendicular to the triangular face.

b = 8, h = 4, H = 10

Substitute the values of the pronumerals into the formula and evaluate. Include the appropriate unit in your answer.

V = × 8 × 4 × 10

V = 160 cm3

112---

2

3 12---

WORKEDExample

10 cm

8 cm

4 cm

199

SkillSHEETanswersTry theseFind the volume of the triangular prisms shown.

1 2

V = × b × h × H V = ..........................................

b = ......., h = ......., H = 12 b = ......., h = ......., H = .......

V = × ....... × ....... × ....... V = ..........................................

V = .......................................... V = ..........................................

3 4

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

5 6

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

8 cm

12 cm

6.9 cm

30 cm10 cm

8.7 cm

12---

12---

52 mm

60 mm95 mm7 m

10 m

6 m

8 m

12 m 13 m

10 m

18 mm24 mm

20 mm

200

SkillSHEETanswersSkillSHEET 9.14Volume of cubes and rectangular prismsThe volume of a prism can be calculated using the formula:

Volume = area of base × heightFor a cube of length l, the area of the base is l2.

So the volume of a cube is given by the formula: V = l3, where l is the length of the side of the cube.Similarly, the volume of a rectangular prism is given by the formula: V = l × w × h, where l is the length,

w is the width and h is the height of the prism.

Try theseFind the volume of each of the following solids.

1 2

V = l3 V = l × w × h

l = . . . . . . . . . . l = . . . . . . . . . . , w = . . . . . . . . . . , h = . . . . . . . . . .

V = . . . . . . . . . . V = . . . . . . . . . . × . . . . . . . . . . × . . . . . . . . . .

V = . . . . . . . . . . cm3 V = . . . . . . . . . . m3

Find the volumes of the following solids.a b

THINK WRITE

a This solid is a cube. Write the formula for the volume of a cube.

a V = l3

Identify the value of l. l = 10Substitute 10 for l into the formula and evaluate.

V = 103

V = 1000 cm3

b The solid shown is a rectangular prism. Write the appropriate volume formula.

b V = l × w × h

Identify the values of the pronumerals. l = 12, w = 8, h = 5Substitute the values of l, w and h into the formula and evaluate.

V = 12 × 8 × 5 V = 480 cm3

10 cm

12 cm

5 cm

8 cm

1

23

1

23

WORKEDExample

2.4 cm12 m

4 m

4 m

201


V = . . . . . . . . . . V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

l = . . . . . . . . . . l = . . . . . . . . . . , w = . . . . . . . . . . , h = . . . . . . . . . .

V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V = . . . . . . . . . . mm3 V = . . . . . . . . . . cm3

5 6

V = . . . . . . . . . . V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V = . . . . . . . . . . mm3 V = . . . . . . . . . . cm3

7 8

V = . . . . . . . . . . V = . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V = . . . . . . . . . . m3 V = . . . . . . . . . . cm3

18 mm4 cm

3.6 cm

19 cm

92 mm

85 mm

11 mm

1.4 cm

1 cm

10.2 cm

12.5 m

4.7 cm

202

SkillSHEETanswersSkillSHEET 10.1Addition of directed numbersFor the addition of directed numbers follow these rules.1. If both numbers have the same sign, add them together. The answer will have the same sign. 2. If numbers have opposite signs, subtract the smaller number from the larger. The answer will have the

sign of the larger number.


1 6 + −3 2 7 + −12 3 −8 + 4 4 −12 + 13

= 6 . . . . . . . . . . 3 = 7 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 −7 + −11 6 12 + 8 7 −9 + −6 8 −8 + −8

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 −2 + 4 + −5 10 12 + −5 + −1 11 6 + 2 + −8 12 −2 + −3 + −7

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Calculate each of the following.a 7 + -5 b -8 + -2

THINK WRITE

a Numbers have opposite signs, so subtract 5 from 7. Since the larger number (7) is positive, so is the answer.

a 7 + −5= 7 – 5= 2

b Both numbers are negative, so add them together. The answer will have the same sign as the numbers (that is, negative).

b −8 + −2= −8 − 2= –10

WORKEDExample

203

SkillSHEETanswersSkillSHEET 10.2Moving up and down or left and rightSeveral translations in one (either horizontal or vertical) direction can be replaced with a single translationby using the following rules.1. If two translations are in the same direction (for example, both up), add the number of units for each

translation and keep the direction. 2. If two translations are in opposite directions (for example, one left and the other right), subtract the

smaller number of units from the larger and keep the direction that had the larger number of units.

Try theseReplace each of the following with a single translation that would have taken the object from its startingpoint to its final position.

1 2 up, 7 up . . . . . . . . . . . . . . . . . . . 2 3 down, 4 down . . . . . . . . . . . . . . . . . . .

3 5 left, 1 left . . . . . . . . . . . . . . . . . . . 4 3 right, 6 right . . . . . . . . . . . . . . . . . . .

5 4 up, 3 up, 5 up . . . . . . . . . . . . . . . . . . . 6 7 down, 1 down, 3 down . . . . . . . . . . . . . . . . . . .

7 8 right, 2 right, 5 right, 2 right . . . . . . . . . . . . . . . . . . . 8 1 left, 2 left, 3 left, 4 left . . . . . . . . . . . . . . . . . . .

9 3 up, 5 down . . . . . . . . . . . . . . . . . . . 10 4 down, 7 up . . . . . . . . . . . . . . . . . . .

11 8 left, 2 right . . . . . . . . . . . . . . . . . . . 12 6 right, 1 left . . . . . . . . . . . . . . . . . . .

13 2 up, 3 down, 7 up . . . . . . . . . . . . . . . . . . . 14 6 left, 2 left, 9 right . . . . . . . . . . . . . . . . . . .

15 4 up, 7 down, 3 up, 1 down . . . . . . . . . . . . . . . . . . . 16 8 left, 4 right, 7 right, 3 right, 5 left . . . . . . . . . . . . . . . . . . .

Replace each of the following with a single translation that would have taken the object from itsstarting point to its final position.a 3 down, 1 down, 2 down b 4 left, 6 right

THINK WRITE

a Since all translations are in the same direction (that is, all three are down), add the units together.

a 3 + 1 + 2 = 6

Replace three translations down with a single translation in the same direction (that is, down).

3 down, 1 down, 2 down = 6 down

b Since the translations are in opposite directions, subtract the smaller unit from the larger.

b 6 − 4 = 2

Replace the two translations with a single one. Since translation to the right had more units, the final translation must be in the same direction (that is, to the right).

4 left, 6 right = 2 right

1

2

1

2

WORKEDExample

204

SkillSHEETanswersSkillSHEET 10.3Translation of a pointIf a point is moved up, down, right, left or in any combination of these directions, we say that it has beentranslated.

Try theseUse graph paper to show the original position of point P and its position after each of the following trans-lations.

1 2 right 2 3 left 3 5 down

4 2 left, 1 up 5 3 right, 2 up 6 1 left, 4 down

7 2 down, 3 left, 1 down 8 3 up, 1 right, 1 down, 2 right 9 2 down, 1 right, 3 up, 2 left

Use graph paper to show the original position of point P and its position after each of the followingtranslations.a 5 right b 2 left, 3 up

THINK DRAW

a Draw a point on graph paper and label it P. Imagine moving the point 5 units (that is, 5 squares) to the right. Draw the point in its new position and label it P′.

a

b Draw a point on graph paper and label it P. Move the point 2 units to the left and then 3 units up. Draw the image of the point after the translation and label it P′.

b

P'P

P'

P

WORKEDExample

205

SkillSHEETanswersSkillSHEET 10.4Reflection of a pointA reflection is an exact image of an object as seen in a mirror. A reflection is as far behind the mirror as theobject is in front of it.

To find a reflection of a point in a given line, follow these steps.Step 1 Draw a perpendicular line from the point to the mirror line.Step 2 Extend the perpendicular line beyond the mirror line.Step 3 Measure the distance from the point to the mirror line along the perpendicular line. The reflection

is on the other side of the mirror, at the same distance along the line.

Try theseFor each of the following, draw a reflection of the given point in the dotted line.

1 2 3

Draw a reflection of the given point in the dotted line.

THINK DRAW

Draw a line from point P so that it is perpendicular to and extends beyond the dotted line.

Point P is 4 units (that is, 4 squares) from the dotted line when measured along the perpendicular line. To find the position of the reflection, measure out 4 units along the perpendicular on the other side of the dotted line. Label the reflection P′.

P

1

P

2

PP'

WORKEDExample

P P

P

206


7 8 9

10

P

PP

P

P

P

P

207

SkillSHEETanswersSkillSHEET 10.5Reflection of an objectA reflection is an exact image of an object, as seen in a mirror. A reflection is as far behind the mirror as theobject is in front of it.

A reflection always has reversed orientation; that is, right appears left and left appears right. A mirror isuseful for looking at reflections of various shapes.

Draw reflections of each of the following objects using the dotted line as the ‘mirror’ line. a b

THINK DRAW

a A reflection is as far behind the mirror as the shape is in front of it. Use a mirror to help you draw the reflection.

a

b Reflect the shape in the mirror line. Make sure that all vertices of the image are as far from the line as the vertices of the original shape.

b

WORKEDExample

208

SkillSHEETanswersTry these1 Draw reflections of each of the following objects in the dotted line.

a b

c

d

2 For each of the following, indicate where a mirror should be placed to produce the given image.

a b

c d

209

SkillSHEETanswersSkillSHEET 10.6Rotation of a pointThe turning of an object about a certain point is called a rotation. To rotate a point half a turn (180°), followthese steps.

Step 1 Join the given point and the centre of rotation with a straight line.

Step 2 Extend the line beyond the centre of rotation.

Step 3 Locate the image: it is as far from the centre of rotation as the original point.

To rotate a point any number of degrees in either a clockwise or anticlockwise direction, follow these steps.

1. Join the given point with the centre of rotation.

2. Position your protractor so that its centre is at the centre of rotation; that is, the 0 mark is at the givenpoint and angle size increases in the required direction (clockwise or anticlockwise).

3. Measure out the required number of degrees and put a small mark.

4. Join the centre of rotation and the degree mark with a straight line.

5. Measure the distance of the given point from the centre of rotation.

6. Locate the image along the new line: it is as far from the centre of rotation as is the original point.

Show the final position of point P after a half-turn rotation about the dot.

THINK DRAW

Join the point P and the centre of rotation with a straight line and extend the line beyond the centre of rotation.

Measure the distance from point P to the centre of rotation: it is 4 units. The image of point P is on the other side of the centre of rotation, 4 units from it along the line. Label the image P′.

P

1

P

2

P P'

1WORKEDExample

210

SkillSHEETanswers

Try these1 For each of the following, show the final position of point P after a half-turn rotation about the dot.

a b c

d e

Show the image of point P after rotating it 90° in a clockwise direction.

THINK DRAW

Join point P with the centre of rotation. Position your protractor so that its centre is at the centre of rotation, its 0 is at point P and angle size increases in the clockwise direction. Mark the 90° angle.

Remove your protractor and join 90° and the centre of rotation with the straight line.

Measure the distance from point P to the centre of rotation (it is 4 units). The image is located the same distance from the centre along the 90° line. Label the image P′.

P

1

P 90°

2

P

3

P90°

P'

2WORKEDExample

P

P

P

P

P

211

SkillSHEETanswers2 Show the image of point P after the following rotations.

a b c

90° anticlockwise 270° anticlockwise 135° clockwise

d e

turn clockwise turn clockwise

P P P

PP

14--- 3

4---

212

SkillSHEETanswersSkillSHEET 10.7Rotation of an object through 180∞An object turning about a certain point is called a rotation.

To specify the rotation we need to identify the centre of rotation (that is, the point about which the objectis to be rotated). We also need to state the size of the angle of the rotation and whether the object is to beturned in a clockwise or anticlockwise direction.

If we consider a rotation of 180°, it is not necessary to specify the direction as clockwise or anticlock-wise because it would produce the same result. The image obtained after a rotation of 180° with the centreof rotation at the centre or middle of the object is the same as looking at the object upside down.

Try these1 Which of the following letters are the same when you look at them upside down?

2 List which of the remaining letters of the alphabet look the same after a rotation of 180° with the centreof rotation at the middle of the letter.

O S H N B

213

SkillSHEETanswersSkillSHEET 11.1Grid coordinates IThe position of a point or symbol can be accurately described using both a horizontal and vertical label. Consider the grid at right.

With a grid system, each square on the grid can be described using a letter and a number. These are called the coordinates of the point or symbol. The horizontal label is listed first, followed by the vertical label.

Try theseGive the coordinates for the following symbols in the grid above.

1 ❍

The symbol is in the . . . . . . . . . . . . . column and the . . . . . . . . . . . . . row.

Column label is . . . . . . . . . . . . .

Row label is . . . . . . . . . . . . .

Coordinates of the symbol: . . . . . . . . . . . . .

2 ❄


Column label is . . . . . . . . . . . . .

Row label is . . . . . . . . . . . . .


A B C D E

1 ✄ ☎ ✆ ❙ ❖

● ❆ ❄ ❁ ❏

❚ ❘ ◆ ✶ ❉

❍ ✱ ✥ ✲ ✴

✹ ✦ ✪ ✷ ■

2

3

4

5

Give the coordinates of the symbol ✶ in the grid above.

THINK WRITE

Locate the symbol on the grid. The symbol is in the fourth column and the third row of the grid.

Consider the horizontal labels across the top of the grid. Write the label of the column that has the symbol you want.

Column label is D.

Consider the vertical label down the side of the grid. Write the label of the row that has the symbol you want.

Row label is 3.

Write the coordinates for the symbol, listing the horizontal label first.

Coordinates of the symbol: D3

1

2

3

4

WORKEDExample

214

SkillSHEETanswers3 ☎


Column label is . . . . . . . . . . . . .

Row label is . . . . . . . . . . . . .


4 ❉


Column label is . . . . . . . . . . . . .

Row label is . . . . . . . . . . . . .


215

SkillSHEETanswersSkillSHEET 11.2Grid coordinates IIThe position of a point or symbol can be accurately described using both a horizontal and vertical label. Consider the grid at right.

With a grid system, each square on the grid can be described using a letter and a number. These are called the coordinates of the point or symbol. The horizontal label is listed first followed by the vertical label.

Try theseDraw the symbol that is found:

1 at E1 2 at B5 3 at C4

Symbol is . . . . . . . . . . . . . Symbol is . . . . . . . . . . . . . Symbol is . . . . . . . . . . . . .

4 3 squares above D4 5 1 square below B3 6 4 squares to the left of E2

Coordinates of required Coordinates of required Coordinates of required

square: . . . . . . . . . . . . . square: . . . . . . . . . . . . . square: . . . . . . . . . . . . .

Symbol is . . . . . . . . . . . . . Symbol is . . . . . . . . . . . . . Symbol is . . . . . . . . . . . . .

A B C D E

1 ✄ ☎ ✆ ❙ ❖

● ❆ ❄ ❁ ❏

❚ ❘ ◆ ✶ ❉

❍ ✱ ✥ ✲ ✴

✹ ✦ ✪ ✷ ■

2

3

4

5

Draw the symbol that is found:a at E5 b 2 squares to the right of A3.

THINK WRITE

a Identify the square in the grid corresponding to E5.

a E5 is the square in the 5th column and the 5th row of the grid.

Draw the symbol. Symbol is ■.

b Identify the square in the grid corresponding to A3.

b A3 is the square in the 1st column and the 3rd row.

Locate the square in the grid that is 2 squares to the right of A3.

The square that is 2 squares to the right of A3 has the coordinates C3.

Draw the symbol. Symbol is ◆.

1

2

1

2

3

WORKEDExample

216

SkillSHEETanswersSkillSHEET 11.3Plotting coordinate pointsThe position of any point on a Cartesian plane can be defined by two numbers called coordinates. Thecoordinates of a point are written as an ordered pair of numbers in brackets. The first number in the bracketsis the x-coordinate and the second number is the y-coordinate of the point. The coordinates of the point Pare therefore written as P(x, y).

The x-coordinate (first number in brackets) indicates the number of units to the right (if positive) or to theleft (if negative) that the point is from the origin. The y-coordinate (second number in brackets) indicatesthe number of units up (if positive) or down (if negative) that the point is from the origin. For example, tolocate a point with coordinates (2, 5), we need to start at the origin and move 2 units to the right and then5 units up.

Try these1 State the size and the direction of the moves from the origin in order to locate each of the following

points.

a A (2, 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b B (3, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c C (0, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d D (1, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

e E (2, –2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f F (–4, –4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g G (–5, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h H (0, –3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i I (–1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j J (3, –4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Draw a Cartesian plane and show the locations of the points in question 1.

State the size and the direction of the moves from the origin in order to locate each of the followingpoints.a A(2, –1) b B(0, 4) c C(–3, 0)

THINK WRITE

a The first number (2) indicates the number of units to the right; the second number (−1) indicates the number of units down.

a 2 right, 1 down

b The x-coordinate is 0, so there is no move across. The y-coordinate indicates the number of units up.

b 4 up

c The x-coordinate indicates the horizontal movement. Since it is negative, it indicates movement to the left. As the y-coordinate is 0, there is no vertical (up or down) move.

c 3 left

WORKEDExample

217

SkillSHEETanswersSkillSHEET 11.4Substitution into rulesTo substitute a given value of the pronumeral into an algebraic sentence. replace the pronumeral with thatvalue. When all pronumerals have been replaced with numbers, the expression can be evaluated. Order ofoperations must be observed at all times when evaluating.

Try these1 Substitute 5 for x in each of the following rules and then find the value of y.

a y = x + 9 b y = x − 3 c y = 12 + x d y = 25 − x

y = 5 + 9 y = . . . . . . . . . . − 3 y = 12 + . . . . . . . . . . y = . . . . . . . . . . . . . .

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

e y = 4x f y = 7x g y = 3x − 4 h y = 2x + 6

y = 4 × . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

i y = 25 − 3x j y = 11 + 6x

y = 25 − 3 × . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

Substitute 5 for x in each of the following rules and then find the value of y.a y = x + 7b y = 2x - 3

THINK WRITE

a Replace x with the given value (5). The rest of the expression remains unchanged.

a y = 5 + 7

Add 5 and 7 to find the value of y. y = 12

b Substitute 5 for x, remembering that in algebra 2x means 2 × x.

b y = 2 × 5 − 3

To find the value of y, perform the multiplication first, followed by the subtraction.

y = 10 − 3y = 7

1

2

1

2

WORKEDExample

218

SkillSHEETanswers2 Substitute 3 for x in each of the rules in question 1 and hence find the value of y.

a y = x + 9 b y = x − 3 c y = 12 + x d y = 25 − x

y = 3 + 9 y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

e y = 4x f y = 7x g y = 3x − 4 h y = 2x + 6

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

i y = 25 − 3x j y = 11 + 6x

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

y = . . . . . . . . . . . . . . y = . . . . . . . . . . . . . .

219

SkillSHEETanswersSkillSHEET 11.5Completing a table of values for a given ruleTo complete a table of values for a given rule, follow these steps.Step 1 Substitute each value of x into the rule.Step 2 Calculate the corresponding value of y.Step 3 Put the value of y into the appropriate cell of the table by placing it directly below the corresponding

value of x.

Try theseComplete the table of values for each of the following.

1 y = x + 9 2 y = x − 5

Complete the table of values for each of the following.

a y = x + 5 b y = 2x - 3

x –2 –1 0 1 2

y 3 4

x –2 –1 0 1 2

y

THINK WRITE

a The first two columns of the table are already filled in. To complete the table, substitute each value for x in turn into the given rule and find corresponding value of y. When x = 0, y = 0 + 5 = 5When x = 1, y = 1 + 5 = 6When x = 2, y = 2 + 5 = 7

a y = x + 5

b Substitute each given value of x into the rule y = 2x − 3, and find the corresponding value of y. Write these values into the appropriate cells of the table.When x = −2, y = 2 × (−2) − 3 = −4 − 3 = −7 When x = −1, y = 2 × (−1) − 3 = −2 − 3 = −5When x = 0, y = 2 × (0) − 3 = 0 − 3 = −3When x = 1, y = 2 × (1) − 3 = 2 − 3 = −1When x = 2, y = 2 × (2) − 3 = 4 − 3 = 1

b y = 2x − 3

WORKEDExample

x –2 –1 0 1 2

y 3 4 5 6 7

x –2 –1 0 1 2

y –7 –5 –3 –1 1

x –2 –1 0 1 2

y 7 8

x –2 –1 0 1 2

y –7 –4

220

SkillSHEETanswers3 y = x + 12 4 y = 5x

5 y = 2x + 8 6 y = 2x − 10

7 y = −3x + 4 8 y = −x − 7

x –2 –1 0 1 2

y 14

x –2 –1 0 1 2

y –5

x –2 –1 0 1 2

y

x –2 –1 0 1 2

y

x –2 –1 0 1 2

y –2

x –2 –1 0 1 2

y –6

221

SkillSHEETanswersSkillSHEET 11.6Plotting a line using a table of valuesTo plot a line whose equation or rule is given, follow these steps.Step 1 Set up a table of values and use the given rule to fill it in.Step 2 Draw a set of axes.Step 3 In the table, each pair of corresponding values of x and y gives the coordinates of a point on the

line. Plot these points on the set of axes.Step 4 Join the points to form a straight line.Step 5 Write the rule of the line next to its graph.

Draw up a table of values and plot the graph for the following rule.y = x + 1

THINK WRITE/DRAW

Set up a table of values, including some negative and positive values of x, and 0. Fill in the table by substituting the values of x into the given rule and calculating the corresponding value of y.When x = −2, y = −2 + 1 = −1When x = −1, y = −1 + 1 = 0When x = 0, y = 0 + 1 = 1When x = 1, y = 1 + 1 = 2When x = 2, y = 2 + 1 = 3

y = x + 1

Draw a set of axes (Cartesian plane).Each pair of corresponding values of x and y in the table gives coordinates of points belonging to the graph. These points are (−2, −1), (−1, 0), (0, 1), (1, 2) and (2, 3). Plot these points on the set of axes.Join the points to form a straight line and label the graph.

1

x –2 –1 0 1 2

y –1 0 1 2 3

2

x

y

67

45

321

–1–2–3–4–5–6–7

1–2–1–3–4–5 2 3

y = x + 1

4 50

3

4

WORKEDExample

222

SkillSHEETanswersTry theseDraw up a table of values and plot the graph for each of the following rules.

1 y = x − 1 2 y = x + 2

3 y = −2x 4 y = 2x − 3

x –2 –1 0 1 2

y –3 –1

x –2 –1 0 1 2

y

x

y

54321

109876

–1–2–3–4–5–6–7–8–9

–10

1–2–1–3–4–5–7–6–8–9–10 2 3 4 5 6 7 8 9 100 x

y

54321

109876

–1–2–3–4–5–6–7–8–9

–10

1–2–1–3–4–5–7–6–8–9–10 2 3 4 5 6 7 8 9 100

x –2 –1 0 1 2

y

x

y

x

y

54321

109876

–1–2–3–4–5–6–7–8–9

–10

1–2–1–3–4–5–7–6–8–9–10 2 3 4 5 6 7 8 9 100 x

y

54321

109876

–1–2–3–4–5–6–7–8–9

–10

1–2–1–3–4–5–7–6–8–9–10 2 3 4 5 6 7 8 9 100

223

SkillSHEETanswers5 y = −x + 4 6 y = −2x + 1

x

y

x

y

x

y

54321

109876

–1–2–3–4–5–6–7–8–9

–10

1–2–1–3–4–5–7–6–8–9–10 2 3 4 5 6 7 8 9 100 x

y

54321

109876

–1–2–3–4–5–6–7–8–9

–10

1–2–1–3–4–5–7–6–8–9–10 2 3 4 5 6 7 8 9 100

224

SkillSHEETanswersSkillSHEET 11.7Measuring the rise and the runTo measure the rise and the run for a straight line, follow these steps.Step 1 Select two points on the line. If the line goes through the origin, it is easiest if you select the origin

and any other point. If the line cuts both axes, select the x-intercept and the y-intercept.Step 2 Construct the gradient triangle, so that the two points are the vertices.Step 3 Measure the horizontal distance (that is, the distance along the horizontal side of a triangle)

between the two points. This distance represents the run. Note that the run is always positive.Step 4 Measure the vertical distance (that is, the distance along the vertical side of a triangle) between the

two points. This distance represents the rise. Note that if the line slopes upward from left to right,the rise is positive, and if the line slopes downward, the rise is negative.

State the rise and the run for each of the following straight lines.a b

THINK WRITE/DRAWa Since the line goes through the origin, select the origin and

some other point. Draw the gradient triangle so that the selected points are the vertices.

a

Run = 2Measure the distance along the horizontal side of the triangle (that is, how much it is from 0 to 2). Hence state the value of the run.Measure the distance along the vertical side of the triangle (that is, how much it is from 0 to 3) to find the value of the rise. Since the line slopes upward from left to right, the rise is positive.

Rise = 3

b Since the line cuts both axes, select the x- and y-intercepts. Draw the gradient triangle so that the selected points are the vertices.

b

Measure the distance along the horizontal side of the triangle (it is from 0 to 6), and hence state the value of the run. Run = 6Measure the distance along the vertical side of the triangle (that is, from 0 to 2) to find the value of the rise. Since the line slopes downward from left to right, the rise is negative.

Rise = −2

y

x

4321

–1–2–3–4

–1 1 2 3 4–2–3–4

y

x

4

21

3

–2–3

–1

–4

2 31 4 5 6–2–3 –1

1y

x

4321

–1–2–3–4

–1 1 2 3 4–2–3–4

23

2

3

1

y

x

4

2–2

61

3

–2–3

–1

–4

2 31 4 5 6–2–3 –1

2

3

WORKEDExample

225

SkillSHEETanswersTry theseState the rise and the run for each of the following straight lines.

1 2

3 4

5 6

y

x

4

2

1

3

–2

–3

–12 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

y

x

4

21

3

–2–3

–1 2 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

y

x

45

21

3

–2–3

–12 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

y

x

4

21

3

–2–3

–1 2 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

y

x

4

21

3

–2–3

–1 2 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

y

x

4

2

1

3

–2–3

–1 2 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

226


9 10

y

x

4

21

3

–2–3

–12 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

y

x

4

21

3

–2–3

–12 31 4 5 6–2–3 –1

Run = .....................Rise = .....................

y

x

4

21

3

–2–3

–1 2 31–2–3 –1

Run = .....................Rise = .....................

y

x

21

3

–2

–4–3

–1 21–2–4 –3 –1

Run = .....................Rise = .....................

227

SkillSHEETanswersSkillSHEET 11.8Substituting 0 for x and y into a ruleZero can be substituted into a rule connecting two variables x and y for either of the variables. When zerois substituted for x, y can be evaluated. If we replace y with zero, the value of x can be found.

Try theseFor each of the following expressions, substitute:

a x = 0 and find the corresponding value of yb y = 0 and find the corresponding value of x.

1 y = x − 3 2 y = x + 5

a When x = 0, y = . . . . . . . . . . – 3 a When x = 0, y = . . . . . . . . . .

When x = 0, y = . . . . . . . . . . When x = 0, y = . . . . . . . . . .

b When y = 0, . . . . . . . . . . = x – 3 b When y = . . . . . . . . . . , . . . . . . . . . . = x + 5

When y . . . . . . . . . . + 3 = x – 3 + 3 . . . . . . . . . . – . . . . . . . . . . = x + 5 . . . . . . . . . .

When y . . . . . . . . . . = x (or x = . . . . . . . . . . ) . . . . . . . . . . = x (or x = . . . . . . . . . . )

3 y = x − 9 4 y = x + 7

a When x = 0, y = . . . . . . . . . . . . . . . . . . . . a When . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

When x = 0, y = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b When y = 0, . . . . . . . . . . = x – 3 b When . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . + . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . = x (or x = . . . . . . . . . . ) . . . . . . . . . . = x (or x = . . . . . . . . . . )

For the expression y = x − 5, substitute:a x = 0 and find the corresponding value of y b y = 0 and find the corresponding value of x.

THINK WRITE

Write the rule. y = x − 5Replace x with zero. a When x = 0, y = 0 − 5Evaluate y. y = −5Now replace y with zero. b When y = 0,

0 = x − 5Add 5 to both sides to find the value of x. (Note that 5 = x and x = 5 are equivalent statements.)

0 + 5 = x − 5 + 55 = xx = 5

1234

5

WORKEDExample

228

SkillSHEETanswers5 y = x − 8 6 y = x + 6

a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 y = −x − 5 8 y = −x + 5

a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 y = −x − 4 10 y = −x + 10

a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229

SkillSHEETanswersSkillSHEET 12.1Reading column graphsGraphical representation of data allows us to see the ‘whole picture at a glance’. Many questions about thedata can be easily answered by simply looking at the graph.

The graph at right shows the number of moccasins manufactured at a small factory over the past year.a During what month was the level of production

at its lowest?b In which months was the level of production

equal?c What was the highest number of moccasins

produced and in what month(s) did this occur?d By how much did the July production exceed

the December production?e Describe and explain any patterns that you can

see.

THINK WRITE

a Find the shortest column and write the answer. a The lowest level of production was in January.

b Look for the columns of equal length (this is best done by using a ruler).

b The level of production was the same in Februaryand November and also in April and October.

c Find the longest column and read its length, using the vertical axis (the frequency).

c In July, the production level achieved its highest —1000 pairs per month.

d Find the number of moccasins produced in December and subtract it from the number produced in July (this is known from the previous question).

d July production = 1000December production = 150So July production exceeds the December production by 1000 − 150 = 850 pairs

e Study the graph and see if it suggests any pattern. Briefly describe and try to explain what you see.

e From the graph the production of moccasins increases steadily from January until July (when it reaches its maximum). It then steadily slows down. The lowest level of production was in January, fol-lowed closely by February. This could probably be explained by the seasonal demand in moccasins, that is, in winter the demand increases dramati-cally, as compared to summer months.

1000900800700600500400300200100

0

Jan

NovOct

Sept

Aug

July

Jun

May

Apr

Mar

Feb

Dec

Production of moccasins

Months

WORKEDExample

230

SkillSHEETanswersTry theseThe graph at right shows monthly car sales for a local caryard over the past year.

1 During what month were the lowest sales figures recorded?

2 In which months was the number of cars sold equal?

3 What was the highest number of cars sold and in what month did this occur?

4 By how much did the December sales exceed the January sales?

5 What was the difference between the highest and the second highest sales figures recorded over the lastyear?

6 What were the total sales for the year?

7 By how much did winter sales exceed the autumn sales?

8 Is there any pattern in sales that you can see?

110100

8070

90

605040302010

Dec

NovOct

Sept

AugJu

l

Jun

May

Apr

Mar

Feb

Jan

Car sales

No.

of

cars

sol

d

Months

231

SkillSHEETanswersSkillSHEET 12.2Reading scales (How much is each interval worth?)When reading scales it is important to remember that the intervals between the adjacent marks are equal. Tofind the value of each interval, find the value of the section of the scale whose endpoints are known (that is,the length between the adjacent major marks) and then divide by the number of intervals along this section.

Try theseFor each of the following scales, find how much each interval is worth.

1 2

50 − 40 = . . . . . . . . . . . . . . 80 − … = . . . . . . . . . . . . . .

10 ÷ … = . . . . . . . . . . . . . . … ÷ … = . . . . . . . . . . . . . .

Each interval is worth . . . . . . . . . . . . . . . . . . . . . . . . . . . . Each interval is worth . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


For each of the following scales, find how much each interval is worth.a b c

THINK WRITE

a Find the value of the section of the scale between the major marks by calculating the difference between the endpoints.

a 30 − 20 = 10

There are 10 intervals between the major marks. So, to find the value of each interval, divide the value of the section of the scale by 10.

10 ÷ 10 = 1

Write the answer in words. Each interval is worth one unit.

b Repeat steps 1–3 as in part a. b 6 − 5 = 11 ÷ 10 = 0.1Each interval is worth 0.1 of a unit.

c Repeat steps 1–3 as in part a. c 200 − 100 = 100100 ÷ 5 = 20Each interval is worth 20 units.

20 30 5 6 100 200

1

2

3

WORKEDExample

40 50 70 80

11 12 23 24

232


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


7 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


9 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


10 20 300 400

8 9 20 30

400 500 4000 5000

233

SkillSHEETanswersSkillSHEET 12.3Reading line graphsLine graphs give meaningful information about the in-between values of particular data. They are com-monly used to display changes over a period of time.

The line graph at right shows the height of a child (Jack) over 5 years.a What was Jack’s height in 2004?b When did Jack’s height reach 150 cm?c How much did Jack grow from 2002 to 2005?

THINK WRITE

a Read vertically up from 2004 on the Years axis to the point intersecting the line graph, then across to the Height axis. Work out the corresponding value on the Height axis and answer the question.

a Jack’s height in 2004 was 156 cm.

b Read horizontally across from the Height axis to the point intersecting the line graph, then down to the Years axis. Work out the corresponding value on the Years axis and answer the question.

b Jack’s height reached 150 cm in 2002.

c State the corresponding height value for 2002 (use your answer to part b).

c Height in 2002 was 150 cm.

Find the corresponding height value for 2005.

Height in 2005 was 163 cm.

Subtract the heights and answer the question.

163 − 150 = 13Change in Jack’s height was 13 cm.

Years

Hei

ght (

cm)

100

110

120

130

140

150

160

170

2001 2002 2003 20052004 2006

Increase in Jack’s heightbetween 2001 and 2006

1

2

3

WORKEDExample

234

SkillSHEETanswersTry these1 The line graph at right represents the

temperature change during a particular day.a What was the temperature at

ii 9.00 am . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii 1.00 pm? . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b At what time/s was the temperature

ii 30°C . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii 25°C? . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c By how much did the temperature increase between 6.00 am and 4.00 pm?

2 The sunrise times on successive Mondays are shown in the following graph.

a What is the sunrise time for Monday of Week 5? . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b When is sunrise earlier than 6.15 am? . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c Between which two weeks does the sunrise time change the most?

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Time of day

Tem

pera

ture

(°C

)

0

5°

10°

15°

20°

25°

30°

35°

0 6 am 7 8 9 10 11 12 1 2 3 4 5 6 pm

Temperature change during the day

Week number

Tim

e of

day

(am

)

6.35

6.30

6.25

6.20

6.15

6.10

6.05

6.00

5.55

5.50

5.45

5.401 2 3 4 5 6 7

Sunrise times

235

SkillSHEETanswersSkillSHEET 12.4Producing a frequency table from a frequency histogramTo produce a frequency table from a frequency histogram, follow these steps.Step 1 Prepare a table with two columns headed ‘Score’ and ‘Frequency’.Step 2 Copy the scores (written underneath the horizontal axis) into the ‘Score’ column.Step 3 The frequency of each score is given by the height of the column above that score, so read the

frequency of each score from the histogram and write it in the corresponding cell of the frequencycolumn.

Copy and complete the following frequency table to show the data represented in the frequency histogram.

THINK WRITE

Consider the frequency of the first score (16). The height of the corresponding column in the frequency histogram is 5. This has already been entered in the table. Consider the next score (17). The frequency of this score is given by the height of the corresponding column, which is 7. So write 7 in the second cell of the frequency column.Read the height of the third column for the third score (18). This corresponds to a frequency of 4, so write 4 in the appropriate cell of the frequency column. Consider the heights of the next three columns (corresponding to the frequencies of the scores 19, 20 and 21). They are 2, 3 and 1 respectively. Write these numbers in the appropriate cells of the frequency column.

Score Frequency

16 5

17

18

19

20

21

321

456789

10

16 18 2017 19 21

Fre

quen

cy

Score

1

2

3

4

WORKEDExample

Score Frequency

16 5

17 7

18 4

19 2

20 3

21 1

236

SkillSHEETanswersTry theseComplete the frequency tables to show the data represented in each frequency histogram.

1

2

3

4

321

456789

10

30 32 3431 33 35

Fre

quen

cy

Score

Score Frequency

30 2

31

32

33

34

35 4

Score Frequency

56 3

57 4

58

59

60

61

62

321

456789

10

56 58 60 6257 59 61

Fre

quen

cy

Score

Score Frequency

78

79

80

81

82 3

321

456789

10

78 80 8279 81

Fre

quen

cy

Score

Score Frequency

100

101

102

103

104

105

3

2

1

4

5

6

7

8

9

10

100 102 104101 103 105

Fre

quen

cy

Score

237

SkillSHEETanswers5

Score Frequency

49

50

51

52

53

54

49 51 5350 52 54

321

456789

10F

requ

ency

Score

238

SkillSHEETanswersSkillSHEET 12.5Presenting data in a frequency tableA set of data can be presented in a frequency table as follows:1. Each possible value of scores in the data set is placed in ascending order in the first column of the table,

named ‘Score’.2. Next, each score in the data set is tallied; that is, one tally mark for every score is placed in the corre-

sponding cell of the ‘Tally’ column.3. Finally, the number of tallies for each score is counted and recorded in the corresponding cell of the

‘Frequency’ column.Note: The frequency of each score shows the number of times each score in the data set has occurred.

Complete the following frequency table for the scores listed below.7, 4, 8, 3, 5, 6, 9, 4, 7, 3, 5, 6, 8, 5, 9, 3, 7, 9, 3, 8, 4, 6, 7, 5, 6

THINK WRITE

Read the first score from the list. It is 7, so put one tally mark in the tally column for the score of 7. Read the next score from the list. It is 4, so put one tally mark for the score of 4. Continue moving along the list and adding one tally mark for each score, until all scores are tallied. Now count the number of tally marks for each score and write that number in the corresponding cell of the frequency column. That is, there are four tally marks for the score of 3, so put 4 in the first cell of the frequency column; there are three tally marks for the score of 4, so write 3 in the corresponding cell of the frequency column and so on, until the table is complete.

Score Tally Frequency

3

4

5

6

7

8

9

1

2

3

4

WORKEDExample


3 4

4 3

5 4

6 4

7 4

8 3

9 3

239

SkillSHEETanswersTry theseComplete the frequency tables for each of the following sets of scores.1 7, 9, 6, 10, 8, 9, 11, 6, 9, 10, 11, 8, 7, 9, 11, 10

2 26, 28, 30, 25, 27, 26, 29, 28, 27, 25, 26, 28, 29, 30, 26, 28, 25

3 22, 21, 17, 19, 20, 18, 23, 21, 17, 23, 19, 18, 20, 22, 21, 18, 23, 20


6

7

8

9

10

11


25

26

27

28

29

30


17

18

19

20

21

22

23

240

SkillSHEETanswers4 35, 33, 39, 37, 34, 36, 38, 33, 36, 35, 34, 38, 37, 36, 39, 34, 36, 37, 35, 35

5 112, 117, 113, 118, 116, 115, 114, 118, 114, 117, 118, 116, 114, 113, 116, 117, 115, 114, 117, 115


33

34

35

36

37

38

39


112

113

114

115

116

117

118

241

SkillSHEETanswersSkillSHEET 12.6Presenting data as a stem-and-leaf plotIn a stem-and-leaf plot the final digit of each number is shown in the leaf column, and the preceding digit(s) is shown in the stem column.

The numbers in both stem and leaf columns are ordered from smallest to largest. The key on the plot letsthe reader know how to read each score.

Try theseFor each of the following sets of data complete the stem-and-leaf plots given.

1 25, 47, 22, 59, 36, 61, 30, 43, 54, 60Key: 22 = 22

Copy and complete the following stem-and-leaf plot of the given data.

34, 56, 49, 61, 37, 70, 63, 52, 41, 61

THINK WRITE

Copy the given stem-and-leaf plot. The key on the plot tells us that the digit in the tens place of each number makes up a stem and the digit in the units place makes up a leaf. Fill in the missing numbers on the plot as follows:i There are two numbers in the set whose stem (that is, the digit in

the tens place) is 3. They are 34 and 37. As 34 is already listed, enter 37 by placing its leaf (7) in the space provided.

ii There are two numbers in the set whose stem is 4. They are 41 and 49. As 49 is already listed, enter 41 by placing its leaf (1) in the space provided.

iii There are two numbers in the set whose stem is 5. They are 52 and 56. Enter these numbers by placing their leaves (2 and 6) in order of increasing size in the spaces provided.

iv There are three numbers in the set whose stem is 6. They are 61, 61 and 63. As one of the scores (61) has already been listed, enter the remaining two numbers by placing their leaves in the space provided.

v Finally, there is one number in the set whose stem is 7. It is 70. Enter this number by placing its leaf (0) in the space provided.

1

2

WORKEDExampleKey: 3 4 = 34

Stem34567

Leaf4 ....... ....... 9....... .......

1 ....... ..............

Key: 3 4 = 34

Stem34567

Leaf4 71 92 61 1 30

Stem23456

Leaf2 .......

....... .......

....... 7

....... .......

....... 1

242

SkillSHEETanswers2 70, 66, 85, 90, 93, 84, 69, 71, 88, 67

Key: 66 = 66

3 101, 122, 117, 135, 104, 112, 139, 140, 143, 126, 129, 137, 136, 125

Key: 101 = 101

4 64, 48, 59, 55, 71, 49, 64, 69, 70, 57

Key: 48 = 48

5 252, 269, 248, 273, 250, 276, 281, 259, 262, 277, 280, 265, 264, 271, 266

Key: 248 = 248

Stem6789

Leaf6 ....... .......

....... .......

....... ....... 8

....... .......

Stem1011121314

Leaf1 .......

....... .......

....... 5 ....... .......

....... ....... ....... 9

....... .......

Stem4567

Leaf8 .......

....... ....... .......

....... 5 .......

....... .......

Stem2425262728

Leaf8....... ....... .......

....... ....... ....... 6 .......

....... ....... ....... .......

0 .......

243

SkillSHEETanswersSkillSHEET 12.7Finding the meanThe mean is the average of all values in a set of data. To find the mean:1. find the sum of all scores in the set2. divide the sum by the number of scores in the set.

Try these

1 a Find the sum of all scores for the following data: 3, 5, 12, 13, 8, 7.b Divide this sum by the number of scores in the data set to find the mean.

a Sum of scores = 3 + ....... + ....... + ....... + 8 + ....... Sum of scores = ..............

b Number of scores = ..............

Mean =

Mean = ….

2 a Find the sum of all scores for the following data: 6, 9, 11, 7, 10, 12, 5, 13, 8.b Divide this sum by the number of scores in the data set to find the mean.

a Sum of scores = 6 + ....... + ....... + ....... + ....... + ....... + ....... + ....... + .......Sum of scores = ..............


Mean =

Mean = ..............

3 a Find the sum of all scores for the following data: 51, 45, 51, 47, 51, 49.b Divide this sum by the number of scores in the data set to find the mean.

a Sum of scores = ................................................................................................................Sum of scores = ..............


Mean =

Mean = ..............

a Find the sum of all scores for the following data: 2, 3, 7, 5, 2, 4, 5.b Divide this sum by the number of scores in the data set to find the mean.

THINK WRITE

a Add all scores in the set together. a Sum of scores = 2 + 3 + 7 + 5 + 2 + 4 + 5Sum of scores = 28

b Count the number of scores in the set. b Number of scores = 7Divide the sum obtained in part a (28) by the number of scores. Mean =

Mean = 4

12 28

7------

WORKEDExample

.......

.......--------

.......

.......--------

.......

.......--------

244

SkillSHEETanswers

4 a Find the sum of all scores for the following data: 7.4, 7.6, 7.9, 7.7, 7.4, 7.6, 7.7, 7.8.b Divide this sum by the number of scores in the data set to find the mean.



Mean =

Mean = ..............

5 a Find the sum of all scores for the following data: 5.8, 4.4, 5.8, 6.2, 5.9, 5.7, 5.5, 4.7.b Divide this sum by the number of scores in the data set to find the mean.



Mean =

Mean = ..............

.......

.......--------

.......

.......--------

245

SkillSHEETanswersSkillSHEET 12.8Arranging a set of data in ascending orderTo arrange a set of numbers in ascending order, rewrite them in order from smallest to largest.

Try theseArrange the following sets of data in ascending order.

1 33, 27, 30, 22, 39, 42, 24

.............., 24, .............., .............., .............., .............., 42

2 81, 59, 68, 74, 65, 77, 86, 60

59, .............., .............., .............., .............., .............., .............., ..............

3 139, 124, 155, 137, 117, 140, 152, 136

.............., .............., .............., .............., .............., .............., .............., ..............

4 300, 284, 290, 292, 276, 311, 293, 307, 289

.............., .............., .............., .............., .............., .............., .............., .............., ..............

5 7.0, 12.6, 7.5, 9.3, 6.9, 10.2, 13.1, 8.4, 9.9

.............., .............., .............., .............., .............., .............., .............., .............., ..............

Arrange the following set of data in ascending order.23, 35, 17, 32, 27, 20, 31

THINK WRITE

Select the smallest number in the set (17) and write it first. The second smallest number is 20, so write it next. Continue to look through the set and select the next smallest number until all numbers are used. It is helpful to cross out each number that you have used from the original list as you go.

17, 20, 23, 27, 31, 32, 35

WORKEDExample

246

SkillSHEETanswersSkillSHEET 12.9Finding the location of the medianIf a set of data contains n scores, the median is given by the th score.

Note that, for the sets of data containing an odd number of scores, the median will be one of the actualscores; for the sets with an even number of scores, the median will be positioned halfway between the twomiddle scores.

Try theseState the location of the median for the sets of data containing the following number of scores.

6 scores 7 scores

1 n = 6 2 n = 7

Location = Location =

Location = Location =

Location = .............. Location = ..............

So the median is So ..........................................

.......................................... ..........................................

.......................................... ..........................................

n 1+2

------------

State the location of the median for a set of data containing:a 5 scores b 12 scores.

THINK WRITE

a Identify the value of n. a n = 5

Substitute the value of n into the formula for the location of the median and simplify.

Location of the median =


Location of the median = 3State the position of the median. So the median is the 3rd score in the set.

b Identify the value of n. b n = 12

Substitute the value of n into the formula

and simplify.



Location of the median = 6

State the position of the median. So the median is halfway between the 6th and the 7th scores in the set.

1

25 1+

2------------

62---

3

1

2

n 1+2

------------

12 1+2

---------------

132------

12---

3

WORKEDExample

....... 1+2

----------------- ....... 1+2

-----------------

.......

2--------- .......

.......--------

247

SkillSHEETanswers3 8 scores 4 9 scores

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

5 10 scores 6 20 scores

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................


.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................


.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

.......................................... ..........................................

248

SkillSHEETanswersSkillSHEET 12.10Finding the middle score of a set of dataThe middle score of a set of numbers is a score such that there is an equal number of scores both to the leftand to the right of it. In order to find the middle score, the set of data must first be arranged in numericalorder.

Note: In this SkillSHEET we will only consider sets with odd numbers of scores.

Try theseFind the middle score for the following sets of numbers by arranging them in numerical order.

1 33, 27, 30, 22, 39, 42, 24

22, 24, .............., .............., .............., .............., ..............

The middle score is ..............

2 81, 59, 68, 74, 65, 77, 86, 60, 52

52, .............., .............., .............., .............., .............., .............., .............., ..............


3 139, 124, 155, 137, 117, 140, 152, 136, 141, 120, 109

.............., .............., .............., .............., .............., .............., .............., .............., .............., .............., ..............


4 300, 284, 290, 292, 276, 311, 293, 307, 289

.............., .............., .............., .............., .............., .............., .............., .............., ..............


5 7.0, 12.6, 7.5, 9.3, 6.9, 10.2, 13.1, 8.4, 9.9

.............., .............., .............., .............., .............., .............., .............., .............., ..............


Find the middle score for the following set of numbers by arranging them in numerical order. 23, 35, 17, 32, 27, 20, 31

THINK WRITE

Rewrite the given set in ascending order. 17, 20, 23, 27, 31, 32, 35Count the number of scores in the set. There are 7 scores. The middle score is the fourth score, as there are 3 scores to the left and to the right of it. Count out the fourth score (from either end) in the set and write the answer.

The middle score is 27.12

WORKEDExample

249

SkillSHEETanswersSkillSHEET 12.11Finding the middle score for data arranged in a stem-and-leaf plotThere is a simple and convenient technique for finding the middle score of a data set that is arranged in astem-and-leaf plot.Step 1 Cross off the first and the last number on the leaf part of the stem-and-leaf plot.Step 2 Cross off the second and the second last number on the leaf part of the stem-and-leaf plot.Step 3 Continue to cross off one number from the top and one number from the bottom of the plot,

remembering to move forward (away from the stem) at the top and backward (towards the stem) atthe bottom of the plot.

Step 4 The last remaining number that is not crossed off is the middle score of the given data set.

Note: In this SkillSHEET we will only consider data sets with an odd number of scores.

Find the middle score of the following data set.Key: 1 | 3 = 13

Stem12345

Leaf3 6 80 3 7 74 6 7 8 93 5 60 1

THINK WRITE

On the leaf part of the stem-and-leaf plot, cross off the first and the last number. That is, cross off 3 and 1.

Key: 1 | 3 = 13

Next cross off the second number and the one before last (6 and 0).

Continue moving towards the middle of the plot, crossing off one number from the top and one number from the bottom, until there is only one number left. This number is the middle score.

State the middle score using the given key. The middle score has a stem that equals 3 and a leaf that equals 6. (Using the given key this should be read as 36.)

The middle score is 36.

1

Stem12345

Leaf3 6 80 3 7 74 6 7 8 93 5 60 1

2

3

4

WORKEDExample

250

SkillSHEETanswersTry theseFind the middle score for each of the following data sets.

1 Key: 1 | 4 = 14 2 Key: 5 | 7 = 57

The middle score is .............. The middle score is ..............

3 Key: 11 | 2 = 112 4 Key: 25 | 7 = 257


5 Key: 10 | 3 = 10.3 6 Key: 16 | 0 = 160


Stem12345

Leaf4 7 80 4 4 50 0 1 3 72 3 61 2

Stem56789

Leaf7 90 1 4 52 3 6 6 8 9 90 1 1 4 52 3 4

Stem1112131415

Leaf2 3 8 91 4 7 7 80 1 1 33 5 96 7 7

Stem2526272829

Leaf70 3 71 3 6 8 92 3 4 5 75 6 8

Stem1011121314

Leaf3 5 51 2 6 81 1 3 94 72 5

Stem1617181920

Leaf0 0 1 4 4 5 70 1 2 3 6 8 81 1 2 2 302

251

SkillSHEETanswersSkillSHEET 12.12Finding the score in a data set that occurs most frequentlyTo find the score in the data set that occurs most frequently:1. count the number of times each score in the data set occurs2. select the score that occurs most often (that is, occurs the largest number of times).

Try theseFor each of the following data sets find the score that occurs most frequently.

1 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8A score of 3: .............. timesA score of 4: .............. timesA score of 5: .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. times

The score that occurs most frequently is ...........................

2 12, 12, 13, 14, 14, 17, 17, 17, 18A score of 12: .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. times


3 15, 13, 18, 24, 20, 13, 18, 20, 24, 18, 15A score of 13: .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. times


For the following data set find the score that occurs most frequently.3, 7, 9, 5, 4, 3, 5, 7, 3, 9, 5, 3, 3

THINK WRITE

The scores in the given set take 5 different values: 3, 4, 5, 7 and 9. Count the number of times each value occurs in the set.

A score of 3: 5 timesA score of 4: 1 timeA score of 5: 3 timesA score of 7: 2 timesA score of 9: 2 times

State the score that occurs the largest number of times.

The score that occurs most frequently is 3.

1

2

WORKEDExample

252

SkillSHEETanswers4 59, 52, 56, 60, 57, 57, 59, 60, 52, 52, 56, 56, 60, 57, 59, 52

A score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. times


5 36, 37, 39, 44, 39, 36, 33, 44, 37, 39, 33, 36, 44, 37, 39, 33 A score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. timesA score of .............. : .............. times


253

SkillSHEETanswersSkillSHEET 13.1Understanding chance wordsMost everyday events could be described as either certain, likely,even chance (fifty-fifty), unlikely, or impossible to occur. On the probability scale these can be shown as follows:

Try these

Describe the probability of each of the following events occurring using the words certain, likely, evenchance, unlikely or impossible.

1 There will be a leap year in 2012. . . . . . . . . . . . . . . . . . . . . . . . . . .

2 A card drawn from a standard deck of playing cards will be a picture card. . . . . . . . . . . . . . . . . . . . . . . . . . .

3 A fair die is rolled and the number that appears uppermost is greater than 1. . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Two fair dice are rolled and the sum of the numbers that appear uppermost is 13. . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Tonight there will be a weather update on the 6 o’clock news. . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Half of the students in your class will be late to school tomorrow morning. . . . . . . . . . . . . . . . . . . . . . . . . . .

7 You will be studying mathematics next year. . . . . . . . . . . . . . . . . . . . . . . . . . .

8 A pink marble is drawn from a bag containing 3 red and 4 yellow marbles. . . . . . . . . . . . . . . . . . . . . . . . . . .

9 A red marble is drawn from a bag containing 2 red and 6 green marbles. . . . . . . . . . . . . . . . . . . . . . . . . . .

10 A fair die is rolled twice. The sum of the numbers that appear uppermost will be greater than 1.

Impo

ssib

le

Even

cha

nce

Like

ly

Unl

ikel

y

Cer

tain

0 0.5 1.0

Describe the probability of each of the following events occurring, using the words certain, likely, evenchance, unlikely, or impossible.a You roll a standard die and a number larger than 3 appears uppermost.b You draw a card from a standard deck of playing cards and it is the 11 of spades.

THINK WRITE

a Of the six numbers on a standard die (1, 2, 3, 4, 5 and 6), half are greater than 3. So I am just as likely to obtain a number larger than 3 as I am to obtain a number that is 3 or less.

a There is an even chance that this event will occur.

b In a standard deck the number cards are numbered from 2 to 10, so it is impossible to draw an 11, as there is no such card in any suit

b It is impossible for this event to occur.

WORKEDExample

254

SkillSHEETanswersSkillSHEET 13.2Understanding a deck of playing cardsA standard deck of playing cards consists of 52 cards. All cards are divided into 4 suits. There are two black suits: spades (♠) and clubs (♣), and two red suits; hearts (♥) and diamonds (♦). In each suit there are 13cards including a 2, 3, 4, 5, 6, 7, 8, 9, 10, a jack, a queen, a king and an ace. (Note that there is no 1.) Thejack, queen and king are called picture cards.

Try these

For a standard deck of playing cards, state the number of:

1 black cards . . . . . . . . . . . . . . . . . . . . . . . . . . 2 aces . . . . . . . . . . . . . . . . . . . . . . . . . .

3 picture cards . . . . . . . . . . . . . . . . . . . . . . . . . . 4 queens of hearts . . . . . . . . . . . . . . . . . . . . . . . . . .

5 kings . . . . . . . . . . . . . . . . . . . . . . . . . . 6 clubs . . . . . . . . . . . . . . . . . . . . . . . . . .

7 not spades . . . . . . . . . . . . . . . . . . . . . . . . . . 8 red cards . . . . . . . . . . . . . . . . . . . . . . . . . .

9 tens . . . . . . . . . . . . . . . . . . . . . . . . . . 10 red jacks . . . . . . . . . . . . . . . . . . . . . . . . . .

11 black threes . . . . . . . . . . . . . . . . . . . . . . . . . . 12 red nines . . . . . . . . . . . . . . . . . . . . . . . . . .

13 number cards greater than 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 14 red picture cards . . . . . . . . . . . . . . . . . . . . . . . . . .

KQ

J 10 9 8 7 6 5 4 3 2

2

A

KQ

J 10 9 8 7 6 5 4 3 2A

2

KQ

J 10 9 8 7 6 5 4 3 2

A

2

KQ

J 10 9 8 7 6 5 4 3 2

A

2

For a standard deck of playing cards, state the number of:a diamondsb black queens.

THINK WRITE

a Diamonds is one of the four suits and there are 13 cards in any suit.

a There are 13 diamonds.

b There is one queen in each suit and there are two black suits (clubs and spades).

b There are 2 black queens.

WORKEDExample

255

SkillSHEETanswersSkillSHEET 13.3Forming fractionsFractions are used to represent parts of a whole. They consist of a top number, called the numerator and abottom number, called the denominator. The denominator shows how many objects (items, parts etc.) thereare, while the numerator shows how many we are concerned with. That is, a fraction shows the number ofobjects (items, parts etc.) that we are concerned with out of all available objects. So to form a fraction, weput the number of things we are concerned with over the total number of these things.

Try theseForm fractions to illustrate each of the following. (Note: Do not simplify.)

1 One sector in a sector graph is 120° (that is, out of 360°).

2 There are three blue marbles in a bag of ten.

3 Rachel spent $1 on lunch out of her $5 pocket money.

4 There are 7 girls in a class of 18.

5 In a bag of 20 balloons there are 8 green ones.

6 Nathan broke a chocolate into 4 equal pieces and gave one to his sister.

7 In a deck of 52 cards there are 4 aces.

8 Surprisingly, 70% of all students in the class passed the test.

9 Lena solved 85 problems from the book that contained 100 problems.

10 Alex spends 3 hours a day travelling to and from work.

11 Michael has 3 weeks of holidays every year.

12 Maestro the cat ate 4 goldfish from the tank that contained 10 goldfish.

13 In a deck of 52 cards, 13 are spades.

14 There are 4 queens in a deck of cards; 2 of them are red.

15 There are 13 diamonds in a deck of cards; 3 of them are picture cards.

16 In a sector graph one of the sectors has an angle which measures 45°.

17 In a sector graph one of the sectors represents 40% of the data.

18 Yesterday, 25% of all cars in the car park were white.

Form fractions to illustrate each of the following.a One sector of a sector graph is 45°.b 7 marbles in a pack of 20 are red.

THINK WRITE

a Altogether there are 360° in a circle, so the denominator is 360. The sector that we are concerned with is 45°, so the numerator is 45.

a

b There are 20 marbles in a pack, so the denominator is 20. There are 7 marbles that we are concerned with (that is, red marbles), so the numerator is 7.

b

45360---------

720------

WORKEDExample

256

SkillSHEETanswersSkillSHEET 13.4Simplifying fractionsTo simplify a fraction, divide both numerator and denominator by the highest common factor.


1 = . . . . . . . . . . . .

2 = . . . . . . . . . . . .

3 = . . . . . . . . . . . .

4 = . . . . . . . . . . . .

5 = . . . . . . . . . . . .

6 = . . . . . . . . . . . .

7 = . . . . . . . . . . . .

8 = . . . . . . . . . . . .

9 = . . . . . . . . . . . .

10 =. . . . . . . . . . . .

11 = . . . . . . . . . . . .

12 = . . . . . . . . . . . .

13 = . . . . . . . . . . . .

14 = . . . . . . . . . . . .

15 = . . . . . . . . . . . .

16 = . . . . . . . . . . . .

17 = . . . . . . . . . . . .

18 = . . . . . . . . . . . .


THINK WRITE


=

852------

852------ 2

13------

WORKEDExample

1352------ 26

52------ 4

52------

1252------ 48

52------ 2

6---

46--- 8

10------ 15

20------

1220------ 8

26------ 24

42------

1848------ 7

56------ 12

36------

80360--------- 135

360--------- 99

360---------

257

SkillSHEETanswersSkillSHEET 13.5Converting a fraction into a decimalTo convert any fraction into a decimal, divide its numerator by the denominator. This can be easily doneusing a calculator.

Try theseConvert each of the following fractions into decimals.

1 2 3 4 5

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

6 7 8 9 10

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

Convert into a decimal.

THINK WRITE

To convert the given fraction into a decimal, divide 15 by 80. This can be done with the help of a scientific calculator by pressing the following sequence of buttons and recording the result:

With a graphics calculator, press the keys for

and then finish by pressing or .

= 0.1875

1580------

15 ÷ 80 =

15 ÷ 80 EXEENTER

1580------

WORKEDExample

1540------ 5

16------ 7

8--- 25

80------ 41

100---------

45300--------- 91

128--------- 15

64------ 138

200--------- 159

160---------

258

SkillSHEETanswersSkillSHEET 13.6Converting a fraction into a percentageTo convert a fraction into a percentage, multiply the fraction by 100%.

Try theseConvert each of the following fractions into percentages.

1 2 3

= × 100% = × 100% = × 100%

= × % = × % = × %

= % = % = %

= . . . . . . . . . . . .% = . . . . . . . . . . . .% = . . . . . . . . . . . .%

Convert each of the following fractions into percentages.a b

THINK WRITE

a To change a fraction to a percentage, multiply by 100%.

a = × 100%

Write 100 as a fraction by putting it over 1. = × %


= %

Simplify by dividing 300 by 5. = 60%

b To change a fraction to a percentage, multiply by 100%.

b = × 100%

Write 100 as a fraction by putting it over 1. = × %


= %

Convert the improper fraction into a mixed numeral.

= 66 %

35--- 2

3---

135--- 3

5---

235--- 100

1---------

33005

---------

4

123--- 2

3---

223--- 100

1---------

32003

---------

4 23---

WORKEDExample

34--- 4

5--- 3

8---

34--- 4

5--- 3

8---

34--- 100

.......--------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

259


= × . . . . . . . . . . . .% = × . . . . . . . . . . . .% = × . . . . . . . . . . . .%

= × % = × % = × %

= % = % = %

= . . . . . . . . . . . .% = . . . . . . . . . . . .% = . . . . . . . . . . . .%

7 8 9

= × . . . . . . . . . . . .% = . . . . . . . . . . . . . . . . . . . . . . . .% = . . . . . . . . . . . . . . . . . . . . . . . .%

= . . . . . . . . . . . . . . . . . . . . . . . .% = . . . . . . . . . . . . . . . . . . . . . . . .% = . . . . . . . . . . . . . . . . . . . . . . . .%

= . . . . . . . . . . . .% = . . . . . . . . . . . .% = . . . . . . . . . . . .%

= . . . . . . . . . . . .% = . . . . . . . . . . . .% = . . . . . . . . . . . .%

25--- 7

8--- 1

3---

25--- 7

8--- 1

3---

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

56--- 3

7--- 5

9---

56---

260

SkillSHEETanswersSkillSHEET 13.7Multiplying a fraction by a whole numberTo multiply a fraction by a whole number:1. change the whole number into a fraction by writing it over 12. simplify as much as possible3. multiply the numerators together and the denominators together4. if the answer is an improper fraction, convert it to a mixed number.

Perform each of the following multiplications.a × 360 b × 25

THINK WRITE

a Write the given problem. a × 360


= ×


= ×


= ×


=


= 216

b Write the given problem. b × 25


= ×


= ×


= ×


=


= 11

60100--------- 45

100---------

160100---------

260100--------- 360

1---------

335--- 360

1---------

4 31--- 72

1------

52161

---------

6

145100---------

245100--------- 25

1------

3920------ 25

1------

4 94--- 5

1---

5454------

6 14---

WORKEDExample

261

SkillSHEETanswersTry thesePerform each of the following multiplications.

1 × 36 2 × 58 3 × 20

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

4 × 32 5 × 70 6 × 160

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

7 × 5000 8 × 250 9 × 80

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

10 × 75 11 × 24 12 × 40

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

20100--------- 10

100--------- 35

100---------

.......

100--------- 36

.......-------- 10

.......-------- .......

1-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

75100--------- 5

100--------- 2

100---------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

17100--------- 92

100--------- 45

100---------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

110100--------- 230

100--------- 125

100---------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

262

SkillSHEETanswersSkillSHEET 13.8Listing the sample spaceA sample space is the list of all the possible outcomes obtained from an experiment. It is usually denotedusing the symbol S, with the list of outcomes enclosed in {} braces and separated by commas.

Try theseList the sample space (possible outcomes) for each of the following experiments.

1 A coin is tossed.

S = {Heads, . . . . . . . . . . . . . . . . . . . . . . . . . .}

2 A fair octahedral (8-sided) die is rolled.

S = {1, 2, . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . .}

3 A circular spinner numbered from 1 to 10 is spun.

S = {. . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . .}

4 A marble is selected from a bag containing 3 green, 5 red and 8 yellow marbles.

S = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 A vowel from the English alphabet is randomly selected.....................................................

6 A card is selected from a deck containing picture cards (jack, king, queen) only.....................................................

7 A person is asked to name the month in which he or she was born.....................................................

8 A day of the school week is selected for a Year 9 Maths test.....................................................

9 An even number between 21 and 51 is randomly selected.....................................................

10 A coin is drawn from a purse containing all types of Australian coins.....................................................

List the sample space (possible outcomes) for the following experiment.A fair standard (6-sided) die is rolled.

THINK WRITE

A fair standard 6-sided die has its faces numbered from 1 to 6. When it is rolled, the outcome is the number that appears on the top face. Since any face can appear as the top face, possible outcomes are all numbers from 1 to 6. List these numbers.

S = {1, 2, 3, 4, 5, 6}

WORKEDExample

263

SkillSHEETanswersSkillSHEET 13.9Calculating the angle in a sector graphThere are 360° in a full circle. So to find the size of a sector in degrees, given its size as a fraction of a fullcircle, multiply the fraction by 360°.

To multiply a fraction by 360°:1. change 360 into a fraction by writing it over 12. simplify as much as possible3. multiply the numerators together and the denominators together4. if the answer is an improper fraction, convert it to a mixed number.

Try theseFind the size of each of the following sectors in degrees, given that their size as a fraction of a circle is:

1 2 3

× 360 × 360 × . . . . . . . . . . . .

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

Finding the size of a sector in degrees, if the sector is of a circle.

THINK WRITE

To express a fraction of a circle in degrees, multiply by 360.

× 360

Convert 360 into a fraction by writing it over 1. = ×


= ×


=

Convert the improper fraction into a mixed number (which in this case is actually a whole number) and include the degree sign.

= 216°

35---

135---

235--- 360

1---------

331--- 72

1------

4216

1---------

5

WORKEDExample

25--- 1

3--- 3

10------

25--- 1

3--- 3

10------

25--- 360

.......--------- 1

3--- .......

.......-------- .......

.......-------- .......

.......--------

21--- .......

1-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

264


× . . . . . . . . . . . . × . . . . . . . . . . . . × . . . . . . . . . . . .

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

7 8 9

× . . . . . . . . . . . . × . . . . . . . . . . . . × . . . . . . . . . . . .

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

10 11 12

× . . . . . . . . . . . . × . . . . . . . . . . . . × . . . . . . . . . . . .

= × = × = ×

= × = × = ×

= = =

= . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . .

34--- 5

12------ 2

9---

34--- 5

12------ 2

9---

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

78--- 4

15------ 4

5---

78--- 4

15------ 4

5---

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

16--- 23

36------ 29

60------

16--- 23

36------ 29

60------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......-------- .......

.......--------

265

SkillSHEETanswersSkillSHEET 13.10Multiplying proper fractionsTo multiply fractions:1. simplify as much as possible2. multiply the numerators together and the denominators together.

Try thesePerform each of the following multiplications.

1 × 2 × 3 ×

= × = = ×

= =

4 × 5 × 6 ×

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

7 × 8 × 9 ×

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . .

Perform the following multiplication: × .

THINK WRITE

Write the question. ×


= ×


=

35--- 1

6---

135--- 1

6---

215--- 1

2---

3110------

WORKEDExample

25--- 1

2--- 1

3--- 1

2--- 3

10------ 1

3---

15--- .......

.......-------- .......

6-------- .......

.......-------- .......

.......--------

.......

.......-------- .......

.......--------

34--- 1

2--- 1

2--- 1

2--- 2

3--- 1

2---

78--- 2

3--- 4

5--- 1

4--- 4

9--- 3

4---

266

SkillSHEETanswers10 ×

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

= . . . . . . . . . . . . . . . . . . . . . . . .

18--- 1

4---

267

SkillSHEETanswersSkillSHEET 14.1The degree of a vertexA network consists of objects connected by lines. The objects are known as vertices and the lines as edges.The number of edges to which each vertex is connected is known as the degree of the vertex.

Try theseState the degree of each vertex in each of the networks shown below.

1 2

Degree of A is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of A is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of B is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of B is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of C is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of C is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of D is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of D is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of E is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

State the degree of each vertex in the network shown.

THINK WRITE

Starting with any vertex, count the number of edges. The number of edges is equal to the degree.

Vertex A is connected to B, C and E, so the degree is 3.

Repeat step 1 for all the remaining vertices.

Vertex B is connected to A and C, so the degree is 2.Vertex C is connected to A, B, D and E, so the degree is 4.Vertex D is connected to C and E, so the degree is 2.Vertex E is connected to A, C and D, so the degree is 3.

B

E

DC

A

1

2

WORKEDExample

C

D

A

B

BE

D

C

A

268


Degree of A is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of A is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of B is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of B is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of C is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of C is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of D is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of D is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of E is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of E is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of F is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degree of F is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Degree of A is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of B is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of C is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of D is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of E is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of F is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Degree of G is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C

E

DA

F

BC

E

D

A

F

B

C

G

E

DA F

B

269

SkillSHEETanswersSkillSHEET 14.2Vertices, edges and regions of a networkA network where no edges cross over each other is known as a planar network. In these networks, the edgesdivide the network into regions or faces. The vertices are not a part of any region. The space outside a net-work is always counted as a region.

Try theseFind the number of vertices, edges and regions in each network shown below.

1 2

The number of vertices is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The number of vertices is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The number of edges is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The number of edges is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The number of regions is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The number of regions is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Find the number of vertices, edges and regions in the network shown.

THINK WRITE

Count the number of vertices. There are 5 vertices.Count the number of edges. Take care not to count any edge twice.

There are 7 edges.

Count the number of regions. Remember the region outside the network.

There are 4 regions.

B

E

DC

A

12

3

WORKEDExample

C

D

A

B

BE

D

C

A

270


The number of vertices is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The number of vertices is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The number of edges is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The number of edges is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The number of regions is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The number of regions is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

The number of vertices is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The number of edges is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The number of regions is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C

E

DA

F

BC

E

D

A

F

B

C

G

E

DA F

B

271

SkillSHEETanswersSkillSHEET 14.3Euler’s formula for planar networksEuler’s rule for all planar networks is given by the formula:

V = E − F + 2where V is the number of vertices, E is the number of edges and F is the number of faces (also known asregions).

Try theseConfirm Euler’s formula for each network shown below.

1 2

V = . . . . . . . . , E = . . . . . . . . , F = . . . . . . . . V = . . . . . . . . , E = . . . . . . . . , F = . . . . . . . .

V = E − F + 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Confirm Euler’s formula for the network shown.

THINK WRITE

Count the number of vertices, edges and faces in the network.

V = 5, E = 7, F = 4

Write Euler’s formula. V = E − F + 2Substitute the values into the formula and simplify.

5 = 7 − 4 + 25 = 3 + 25 = 5

Compare both sides of the formula and draw your conclusion.

The right-hand side and left-hand side of the formula are both the same; therefore, Euler’s formula is confirmed.

B

E

DC

A

1

23

4

WORKEDExample

C

D

A

B

BE

D

C

A

272


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SkillSHEETanswersSkillSHEET 14.4Paths in networksA path in a network begins at one vertex and ends at another. A path through a network that uses each edgeexactly once is known as an Euler path. A path that goes through each vertex exactly once is known as aHamiltonian path.

Try theseFor each of the networks below:a determine whether there is an Euler path through the network and, if so, give an exampleb find a Hamiltonian path.

1 2

a Euler path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Euler path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b Hamiltonian path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b Hamiltonian path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Use the network shown to find:a an Euler pathb a Hamiltonian path

THINK WRITE/DRAW

a Determine the degree of each vertex. a There are exactly two odd vertices and three even vertices.

Since there are exactly 2 vertices with odd degrees (A and E), an Euler path has to start and finish there; for example, it could begin at A and end at E. Attempt to find a path that uses each edge.

List the sequence of vertices along the path.

An Euler path is A–C–B–A–E–D–C–E.

b Choose any starting vertex and try to visit all the other vertices of the network. (Not all the edges need to be used.)

b A possible Hamiltonian path is A–B–C–D–E.

B

E

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a Euler path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Euler path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b Hamiltonian path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b Hamiltonian path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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a Euler path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b Hamiltonian path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SkillSHEETanswersSkillSHEET 14.5Traversable networksA network is traversable if it can be drawn without lifting the pen off the paper or going over an edge twice.To decide whether or not a network is traversable, the following information can be used.• A network is traversable (starting and finishing at the same vertex) if the degree of all the vertices is

even.• A network is traversable (starting at one odd vertex and finishing at the other odd vertex) if there are

exactly two odd vertices.• If there is one odd vertex or more than two odd vertices, the network will not be traversable.

State whether or not the following networks are traversable. Give a reason for your answer.a b c

THINK WRITE

a Find the degree of each vertex. a All the vertices have an even degree.Check whether the network is traversable and give a reason.

As all the vertices have an even degree, the network is traversable — it will start and finish at the same vertex. One traversable path is B–A–F–D–B–E–F–C–B.

b Find the degree of each vertex. b Vertices D and E are odd. All the other vertices are even.

Check whether the network is traversable and give a reason.

As there are exactly 2 odd vertices the network is traversable. If we start at vertex D, we will finish at vertex E. One traversable path is D–A–B–D–E–C–B–E.

c Find the degree of each vertex. c Vertex A is even. The other four vertices are odd (all of them have a degree of 3).

Check whether the network is traversable and give a reason.

As there are more than two odd vertices the network is not traversable.

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SkillSHEETanswersTry theseState whether or not each of the following networks is traversable. Give a reason for your answer

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