3

Place Value

Thousandths

Thousands

Hundreds

Tens

Ones

Decim

al P

oin

t

Tenths

Hundredths

1000s 100s 10s 1s .

10

1

100

1

1000

1

Know the meanings of these column headings is very important. It tells us the value of each digit. This can be especially useful when rounding to significant figures, multiplying and dividing by 10,100,1000…, converting between fractions decimals and percentages and long multiplication and division. For example

1 7 6 9 . 8 7 3

This digit is not just worth 7. It has a value of 700.

This digit is not just 9. It is worth 9 ones. he same as nine whole ones or ninety tenths, or 0.9 tens.

This digit is not worth 7. It has a value of 7 thousandths or

1000

7

4

MULTIPLICATION

You must learn your tables Just do it!!

Long Multiplication

GRID METHOD

X 20 8

40 800 320

6 120 48

Example 1 28 X 46 = 1288

Add the numbers up 800 320 120 + 48 1288

Example 2 146 X 39 = 5694

X 100 40

30 3000 1200

9 900 360

6

180

54

Sum of each row

4380

1314

5694

Pitfall: Don’t forget zeros Line up columns for addition

5

CHINESE METHOD

28 X 46 = 1288

Pitfall: Make sure you get the diagonals the right way. This is a quick and reliable method to use

0

8

1

2

3

2

4

8

2 8

4

6

DECOMPOSITION METHOD

Example 1 43 X 6 = 258 Method 40 X 6 = 240 3 X 6 = + 18 258 Example 2 27 X 4 = 108 Method 20 X 4 = 80 7 X 4 = + 28 108

Pitfall: This only works if you are multiplying by one digit (i.e. single digits)

6

DIVISION Method - Bus stop method

Example 1 840 ÷ 5 1 6 8 5 8³44 0 5’s into 8 go 1 remainder 3, carry the 3 5’s into 34 go 6 remainder 4, carry the 4 5’s into 40 go 8 So 840 ÷ 5 = 168

1 0 4 6 6 2²4 6’s into 6 go 1 6’s into 2 go 0 carry the 2 6’s into 24 go 4 So 624 ÷ 6 = 104

Example 2 624 ÷ 6

7

Example 3: 782 ÷ 23 Method Write out the 23 times table up to 9x23 23 46 69 92 115 138 161 184 207 0 3 4 23 7 78 92 23 into 7 goes 0 carry the 7 23 into 78 goes 3 remainder 9, carry the 9 23 into 92 goes 4. So 782 ÷ 23 = 34

Example 4 6175 ÷ 19 Method Write out the 19 times table up to 9x19 19 38 57 76 95 114 133 152 171 0 3 2 5 19 6 61 47 95 19 into 6 goes 0 carry the 6 19 into 61 goes 3 remainder 4, carry the 4 19 into 47 goes 2 remainder 9, carry the 9 19 into 95 goes 5. So 6175 ÷ 19 = 325

8

Multiplying by 10, 100 and 1000

48 x 10 = 480 (Add a zero) 48 x 100 = 4800 (Add 2 zeros) 48 x 1000 = 48000 (Add 3 zeros) 4.856 x 10 = 48.56 (Move the decimal point one place to the right) 4.856 x 100 = 485.6 (Move the decimal point two places to the right) 4.856 x 1000= 4856 (Move the decimal point three places to the right)

Dividing by 10, 100 and 1000

35684 ÷ 10 = 3568.4 (Move the digits one place to the right) 35684 ÷ 100 = 356.84 (Move the digits two places to the right) 35684 ÷ 1000 = 35.684 (Move the digits three places to the right) 35600 ÷ 10 = 3560 (Move the digits one place to the right) 35600 ÷ 100 = 356 (Move the digits two places to the right) 35600 ÷ 1000 = 35.6 (Move the digits three places to the right)

9

BIDMAS

B — Brackets I — Indices D — Division M — Multiplication A — Addition S — Subtraction When you do long calculations you must work them out according to the order of operations. BIDMAS helps you to remember the order.

Example 1 5 + (3 x 8) ÷ 6 Brackets first 5 + 24 ÷ 6 Division 5 + 4 Addition 9

Example 2 3² x (15-7) Brackets first 3² x 8 Indices 9 x 8 Multiplication 72

10

Fractions

Equivalent Fractions

Equivalent fractions have the same value, even though they may look different. These fractions all have the same value:

If you multiply or divide the numerator and denominator by the same number the fraction keeps its value:

18

9

6

3

2

1==

18

9

6

3

2

1==

3×

9×

3×

Finding equivalent fractions:

9×

4

3

12

9

60

45==

Simplifying Fractions:

÷5

÷5

÷15

÷15

Converting Mixed Number and Improper Fractions

Mixed number improper fraction

7

24

7

321

7

33 =

+=

1. Multiply the denominator by the integer

2. Add the product to the numerator

5

27

5

225

5

25 =

+=

For example:

11

Converting Improper Fractions to Mixed Numbers To convert between improper fractions you divide and find the remainder.

5

44

5

24=

44524 remainder=÷

42|5 2

04 r4

Multiplying Fractions

Multiply the numerators and the denominators:

4

3

60

45

6

5

10

9==×

4559 =×

60610 =×

Always simplify

15÷

15÷

Dividing Fractions

When dividing fraction turn the second fraction on it’s head (“Flip it”) and then multiply.

Flip the second fraction

on it’s head then multiply

10

3

20

6

4

3

5

2

3

4

5

2==×=÷

4

3

3

4becomes

632 =×2045 =×

2÷

2÷

12

Multiplying and Dividing Mixed Numbers When multiplying and dividing mixed numbers we first convert them into im-proper (top-heavy) fractions.

9

84

9

44

18

88

6

11

3

8

6

51

3

22 ===×=×

3

8

3

22 =

6

11

6

51 =

1863

88118

=×

=×

84944 remainder=÷

Cross-Cancelling

By looking for common factors we can speed up difficult questions of multiply-ing fractions.

2

122

2

45

1

9

2

5

11

18

4

55

11

71

4

313 ==×=×=×

5

1

9

2

Cancel by a factor of 11 Cancel by a factor of 2

Cancel by a factor of 2 Cancel by a factor of 11

Multiplying Fractions and Integers When multiplying fractions and integers change the integer (whole number) into a fraction with a denominator of 1.

4

33

4

15

1

5

4

35

4

3==×=×

1

55 =

13

Adding and Subtracting Fractions To add and subtract fractions you have to use your equivalent fraction skills. When fractions have the same denominator (bottom numbers) we can add the numerators (top numbers).

12

71

24

141

24

38

24

20

24

18

6

5

4

3===+=+

24

18

4

3=

24

20

6

5=

When fractions have the same denominator

we can add the numerators.

Simplify

Adding and Subtracting Mixed Numbers: Method 1: The first method for adding and subtracting mixed numbers is to change first into improper fractions.

15

22

15

32

15

25

15

57

3

5

5

19

3

21

5

43 ==−=−=−

5

19

5

43 =

3

5

3

21 =

15

57

5

19=

15

25

3

5=

322557 =−

14

Adding and Subtracting Mixed Numbers: Method 2: Another method of adding and subtracting mixed numbers is to add and sub-tract the whole numbers (integers) and fractions separately. This method is quicker, but it can be confusing when you combine the whole numbers and fractions together at the end.

It can be confusing when the fractions total is an improper fraction (top-heavy) or a negative number.

15

22

15

22

15

2

15

10

15

12

3

2

5

4

213

3

21

5

43

=+

=−=−

=−

−Whole numbers

(integers)

Fractions

Combine whole numbers

and fractions

Final answer

It can be confusing...

15

Non Calculator Finding the Percentage of a Value

1% Divide by 100

2% Divide by 100 and double it

5% Find 10% and halve it

21/2% Halve 5%

6% Calculate 5% and 1% and add

10% Divide by 10

15% Find 10% and 5% and add

20% Find 10% and double it

25% Halve and halve again (or divide by 4)

30% Find 10% and multiply by 3

50% Halve it

75% Find 50% and 25% and add

Example 1

Find 32% of £125 10% = 12.50, 30% = 3 x 12.50 = 37.50 1% = 1.25, 2% = 2 x 1.25 = 2.50 + 40.00 Note for money there must be 2 decimal places (or none) so the acceptable answers are £40.00 or £40.

Example 2

Find 121/2 % of 16 10% = 1.6, 5% = 0.8 21/2 % = 0.4 121/2 % = 10% + 2

1/2 %, so 121/2 = 1.6 + 0.4 = 2

Non Calculator Finding Percentage Increase or Decrease

The Glossary contains vocabulary associated with increase and decrease.

Example 1

Increase £125 by 32%. From example 1 ‘Finding a Percentage’ find 32% of £125 (£40). Increased amount is £125 + £40 = 165

Example 2

Decrease 16 by 121/2% From example 2 ‘Finding a Percentage’ find 121/2% of 16 (2) Decreased amount is 16 - 2 = 14

16

Calculator Method

for

Finding the Percentage Of, and Increasing or Decreasing a Value

Method uses decimal multipliers . To change a percentage to decimal see page on changing fractions/decimals and percentages.

Finding a Percentage of a Value

Change percentage into decimal by dividing by 100 Multiply decimal by the value.

Example 1

Find 27% of 458 27% = 0.27 27% of 458 = 0.27 x 458 = 123.66

Example 2

Find 81.4% of 34.7 81.4% = 0.814 81.4% of 34.7 = 0.814 x 34.7 = 28.2458

Finding a Percentage Increase or Decrease

If the cost of downloading songs from iTunes was 52p what will it be if it increases by 6% ?

Calculation is 52 x 1.06 = 55.12 because this is money we must round to the nearest penny, 55p

In percentage questions, the first event in time is 100%

First event, 52 is 100%

Second event is 6% more, so 106%

The decimal multiplier for 106% is 1.06 (divide by 100)

17

First event, 125 is 100% Second event is 40% less, so 60%

The decimal multiplier for 60% is 0.60 (divide by 100)

The method works for decrease as well. If a £125 pair of trainers is 40% off in the sale what will be the sale price? Calculation is £125 x 0.6 = £75

18

Changing between Fractions, Decimals and Percentages

Fraction

Percentage Decimal

put

the p

erce

ntage

as

num

erat

or an

d

100 a

s de

nom

inat

or an

d si

mpl

ify

divide the numerator by the denom

inator

multiply by 100

divide by 100

The

per

cent

age

is t

he n

umer

ator

val

ue o

f th

e

equi

vale

nt f

ract

ion

wit

h a

deno

min

ator

of

100

the digits are the numerator and the final

column heading is the denom

inator

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 1

a. 34% = simplifies to

b. 6.4% = simplifies to and this simplifies to

100

34

50

17

100

4.6

1000

64

125

8

19

Example 2

a. The fraction makes which is 84%

b. The fraction makes which is 87.5%

25

21

100

84x4

x4

8

7

x12.5

x12.5

100

5.87

Example 3

a. 0 . 3 4 5 the final column is thousandths so

b. 0 . 0 2 the final column is hundredths so

1/10 1/100

1/1000

1000

345

100

2

Example 4

a. the line in a fraction means divide so means 5 ÷ 8. Use bus-stop or a calculator to get 0.625

b. when the denominator is 10, 100 or 1000 etc it is easy to do the division in your head. So = 3 ÷ 10 = 0.3, = 67 ÷ 100 = 0.67 and = 0.015

8

5

10

3

100

67

1000

15

Example 5 and 6

Multiplying by 100 the digits move two places to the left and dividing by 100 the digits move two places to the right a. so 0.71 becomes 71% and 2.67 becomes 267%

b. Harder examples are 0.6 becomes 60% and 0.356 becomes 35.6%

20

Prime Factors

The Prime Numbers up to 20:

2, 3, 5, 7, 11, 13, 17 and 19

Q. Find the Prime Factors of 420 First draw a Prime Factor Tree

6

42

420

210 2

5

3 2

7

Find 2 numbers which multiply to make 420

Then 210

Then 42

Then 6

The Prime Factors of 420 are 2, 2, 3 ,5, 7

They can be expressed in index form as 2² x 3 x 5 x 7

Finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using Prime Factor trees

Q. Find the LCM and HCF of 12 and 20

12

6 2

3 2

Index form = 2² x 3

20

10 2

5 2

Index form = 2² x 5

21

Put these figures into a Venn Diagram

12 20

3 5 2

2

HCF = 2 x 2 = 4 (Multiply the numbers in the overlapping segment)

LCM = 3 x 2 x 2 x 5 = 60 (Multiply all the numbers)

The Ladder Method

Q. Find the LCM and HCF of 12 and 20

2 12

2 6

3

20

10

5

1. The numbers that

we are using go at the

top of the ladder 2. We write a common

factor of both numbers

beside them.

3. We keep writing

common factors down

as long as possible

To find the HCF multiply the numbers on the left side of the ladder: - HCF = 2 × 2 = 4 To find the LCM multiply the numbers on the left side and the numbers beneath the ladder: - LCM = 2 × 2 × 3 × 5 = 60

22

Rounding Numbers

Rounding to the nearest whole number: Draw a vertical line to the right of the number that will be rounded.

3 . 7 3 8

This number will either stay as a 3 or round up to 4.

Ignore these numbers

The number directly to the right of the vertical line is the decider.

• If the decider is less than 5 the number left of the line stays the same.

• If the decider is 5 or more the number left of the line is rounded up.

So 3.738 rounded to the nearest whole number is 4.

Rounding to the nearest 10

3 4 8 7 The number 7 is the decider. As it is larger than 5 the number to the left of the vertical line will round up.

The answer will either be 3480 or 3490

So 3487 to the nearest 10 is 3490

Rounding to the nearest 100

The answer will either be 3400 or 3500

The number 8 is the decider. It is greater than 5 so 3487 rounded to the nearest hundred is 3500

3 4 8 7

23

Rounding to decimal places

8 9 . 4 8 2

1st decimal place

2nd decimal place

3rd decimal place

Round to 1 decimal place

8 9 . 4 8 2

1st decimal place

The decider

The decider is greater than 5 so the number to the left of the vertical line rounds up

= 89.5

Rounding to significant figures

5 8 4 3 2

1st

2nd 3rd 4th

5th

Significant figures (S.F)

Round to 1 significant figure

5 8 4 3 2

1st S.F The decider

The decider is greater than 5 so 58432 rounded to 1 significant figure = 60000

24

Round to 3 significant figures

5 8 4 3 2

3rd S.F

The decider

The decider is less than 5 so the number to the left of the vertical line stays the same. To round to 3 significant figures we round 58432 down to 58400.

The first significant figure is always the first non zero number so in the number

0 . 0 0 8 3 7 2

the 8 is the first significant figure. A zero counts as a significant figure if it is between two non zero numbers.

Rounding decimals to 2 significant figures

0 . 0 0 8 3 7 2

2nd S.F

The decider

The decider is greater than 5 so 0.008372 is rounded up to 0.0084. When you write out a decimal to significant figures you never add extra zeros to the end of the number. Always stop after the last significant figure.

Remember

Less than 5 stays the same, 5 or more round up!

25

Ratio

Ratio is the share of something in proportion to other shares. For example, if Andy and Graham share sweets in the ratio 1:2, for every sweet Andy gets, Graham gets 2. Ratio can be simplified in the same way fractions can - by finding common factors between all parts. Example

6:12:24

Method 1 6:12:24

All parts have 2 as a common factor, so divide each part by 2

3:6:12 All parts still have 3 as a common factor

so divide each part by 3 1:2:4

All parts only have 1 as a common factor and therefore this is the simplest form.

Method 2 6:12:24

All parts have 6 as a common factor, so divide each part by 6

1:2:4 All parts only have 1 as a common factor and therefore this is the simplest form.

It is O.K. to take more than 1 step if you cant find the highest common factor.

26

Example 1 Divide £360 into this ratio:

Danielle : Emmanuel : Freddie 5 : 4 : 3

In total there are 12 parts (5 + 4 + 3 = 12).

First calculate one part of the ratio: 360 ÷ 12 = £30

Danielle gets 5 parts, so she gets 5 × 30 = £150 Emmanuel gets 4 parts, so he gets 4 × 30 = £120 Freddie gets 3 parts so he gets 3 × 30 = £90

So the money is shared out as follows: D : E : F 5 : 4 : 3 £150 : £120 : £90

If we divide £360 into 12 equal parts of £30, here we see that Danielle takes 5 parts, Emmanuel takes 4 parts and Freddie takes 3 parts.

1

£30

2

£30

3

£30

4

£30

5

£30

6

£30

7

£30

8

£30

9

£30

10

£30

11

£30

12

£30

Danielle

Emmanuel Freddie

27

Example 2 Danielle, Emmanuel and Freddie share some money in the following ratio.

Danielle : Emmanuel : Freddie 5 : 4 : 3

If Freddie gets £18, how much do Danielle and Emmanuel get and how much is there altogether? Note: Notice the difference between Example 1 and Example 2. In Example 1 we are given the total amount, but in Example 2 we are given one persons share. Freddie Gets 3 parts worth £18 in total. We first need to find the value of on share. To find 1 share we would do 18÷3. This equals £6. If one share is £6 then to find out Danielle’s share we need to do 1 share multiplied by 5 - so 6×5. This equals £30. If one share is £6 then to find out Emmanuel’s share we need to do 1 share multiplied by 4 - so 6×4. This equals £24. So … Danielle gets £30 Emmanuel gets £24 Freddie gets £18 In total that is 30+24+18. This equals £72.

1

£6

2

£6

3

£6

4

£6

5

£6

6

£6

7

£6

8

£6

9

£6

10

£6

11

£6

12

£6

Danielle

Emmanuel Freddie