AFM7000 Student Workbook

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Introduction

In the Mathematics L7 module you will learn about the following

things:

Positive and negative numbers

Integers and decimals

Fundamental mathematical operations Exponents

Perfect squares and square roots

Order of mathematical operations

Problem solving questions in the real world

Mathematics is the basis for all science – it will be important in

future studies in engineering and health science.

Use this workbook to help you understand these concepts. Completeall task and answer all questions during the course.





And even when we're relaxing, and playing games maths is necessary

to keep the score and work out who has won!

So maths is used extensively in everyday life – in college, in

workplaces and in our leisure time, which means a great deal depends

on it.

Think of some of the ways that mathematics is used in your

everyday life and list them in the box below:



Positive and

Negative NumberThe basis of mathematics is our numbering system

Positive (+ve): a number greater than zero.

It can be written with or without the plus sign (+).

Example: Positive 3 is written as +3 or 3.

Negative (−ve): a number smaller than zero.

It is indicated with a minus sign (−).

Example: Negative 4 is written as −4.

The positive and negative numbers can be represented on a hori-

zontal number line, as shown below.

If a signed number is a whole number

(for example, 2), it is called an integer.

If a signed number has decimal points

(for example, 2.74) it is called a decimal number.

Integers and Decimals





Examples:

5–2=3, and 15−4=11

−5+2=−3, and –10+3=–7

Let’s look at using the number line to do this

When we add a +ve number, we move in the +ve direction

(forwards)

When we add a −ve number, we move in the −ve direction

(backwards)

Example: Work out 1+2−5. This is (+1)+(+2)+(−5).

We start from 0.

(+1) is +ve so we move forwards by 1 unit and reach +1

(+2) is +ve so we move forward by 2 units and reach +3.

(−5) is −ve so we move backwards by 5 units and reach −2.

Using the rules, gives similarly 1+2−

5=3−

5=−

2.



Subtracting IntegersSubtracting two numbers gives their difference.

To subtract two integers, first we follow the rules –(+)=(−) and –(−)=

(+), and then we add the numbers as explained above.

Examples: (+5)–(+2)=5–2=3; –3–(–5)= –3+5=2.

When we add or subtract more than two integers, we work from

left to right.

Example: −2−(−10)+20=−2+10+20=8+20=28

Multiplication andDivision of IntegersMultiplying two numbers gives their product.

Dividing two numbers gives their quotient.

(Take the division 20¸4=5. The remainder is 0 so we say that 20 is

divided exactly by 4 or 20 divides by 4 or 4 divides into 20).

When we multiply or divide two signed numbers we may omit thebrackets around +ve numbers but we must keep the brackets

around −ve numbers.



Example

(+2)×(−3) can be written as 2×(−3) but not as 2×−3.

When we multiply or divide two integers, we get:

a positive result, if the integers have the same sign

a negative result, if the integers have different signs.

That is, (+)×(+)=(+) (+)÷(+)=(+)

(−)×(−)=(+) (−)÷(−)=(+)

(+)×(−)=(−) (+)÷(−)=(−)

(−)×(+)=(−) (−)÷(+)=(−)

Examples:

3×2=6, (−3)×(−2)=6, 3×(−2)=−6, (−3)×2=−6.

10÷2=5, (−10)÷(−2)=5, 10÷(−2)=−5, (−10)÷2=−5.

You’re Turn!

1. Answer the following Questions:

A. (+2) X (-4)=.................................... B. (-3) X (-5)=.....................................

C. (+3) X (+5)=..................................... D. (+6) ÷ (+3)=.....................................

E. (-8) ÷ (+4)=...................................... F. (-12) ÷ (-3)=....................................

G. (+12) ÷ (+4)=...................................



Three tens 14

Double 7 9

The sum of 26 and 9 30

Half of 18 35

The difference between 22 and 5 17

2. Find the missing number:

A. (-3) X ( ) = (+12) B. ( ) X (-4) = (+20)

C. (+16) ÷ ( ) = (-2)

3. Work out the product of:

A. -6 and -6

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

B. -4 and +9

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

C. +3 and -2

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

4. Draw all the missing lines:

5. Look at these numbers. Draw a ring around the numbers

are multiples of 10:

1000 20 32 70 416325 100 25 26



Adding and

Subtracting DecimalsIf you know how to add and subtract whole numbers, then you can

add and subtract decimals! Let's look at an example.

To add these numbers, first arrange the terms vertically, aligning

the decimal points in each term. Don't forget, for a whole numberlike the first term, the decimal point lies just to the right of the

ones column. You can add zeroes to the right of the decimal point

to make it easier to align the columns. Then add the columns

working from the right to the left, positioning the decimal point in

the answer directly under the decimal points in the terms.

Here's a subtraction example:

To subtract these numbers, first arrange the terms vertically,

aligning the decimal points in each term. You can add zeroes to the

right of the decimal point, to make it easier to align the columns.

Then subtract the columns working from the right to the left,

putting the decimal point in the answer directly underneath the

decimal points in the terms. Check your answer by adding it to the

second term and making sure it equals the first.



Time for some practice!

Try these problems:

1.

A. 0.4 + 0.1 = B. 0.8 + 0.4 =

C. 0.6 + 0.2 = D. 0.8 + 0.5 =

E. 0.6 + 0.9 =

2.

A. 0.48 + 0.99 = B. 0.54 + 0.74 =

C. 0.96 + 0.7 = D. 0.77 + 0.2 =

E. 0.89 + 0.79 =

3.

A. 5.14 + 26.3 = B. 6.91 + 63.9 =C. 1.76 + 37.7 = D. 3.25 + 30.9 =

E. 6.58 + 43.7 =

4.

A. 0.59 - 0.14 = B. 0.88 - 0.34 =C. 0.82 - 0.33 = D. 0.96 - 0.25 =

E. 0.86 - 0.16 =

5.

A. 4.2 - 1.26 = B. 41.8 - 2.78 =

C. 72.2 - 4.95 = D. 86.7 + 9.67 =

E. 62.9 - 8.79 =



Multiplying

Decimals NumbersMultiplying decimals is almost the same as multiplying with whole

numbers. The only difference is that the number of decimal places

in the answer must be the same as the total number of decimal

places in the question.




Try these problems:

1.

A. 0.8 x 7 = B. 0.5 x 7 =

C. 0.1 x 6 =` D. 0.6 x 4 =

E. 0.3 x 3 =

2.

A. 0.2 x 0.5 = B. 0.4 x 0.7 =

C. 0.8 x 0.1 = D. 0.9 x 0.9 =

E. 0.6 x 0.1 =

3.

A. 1.9 x 0.3 = B. 1.6 x 0.5 =

C. 1.6 x 0.5 = D. 1.7 x 0.2 =

E. 1.3 x 0.7 =

4.

A. 5.4 x 0.11 = B. 5.2 x 0.97 =

C. 8.3 x 0.73 = D. 4.6 x 0.11 =

E. 8.2 x 0.75 =



Dividing

Decimals NumbersIf we want to divide two numbers, and one or both of them are

decimal numbers, then:

first we multiply both of them with the same power of 10 (i.e. 10,

100, 1000 …) to make both of them whole numbers, and then we

divide the two whole numbers.

Divide 20 by 0.4

Answer:

Dividing 20 by 0.4 gives the fraction 20/0.4

The numerator is a whole number. The denominator has 1 d.p. so to

convert it to a whole number we need to multiply it by 101=10.

If we have a fraction, we can multiply both the numerator anddenominator by the same number. So in this case we multiply both

20 and 0.4 by 10. This gives

200/4=50 and so 20/0.4=50



Worked example

What is 450 ÷ 0.25?

Method 1

450 ÷ 0.25 =

= Multiply numerator and denominator by 100.

=

= 1800

Method 2

450 ÷ 0.25 = = Multiply numerator and denominator by 4.


Try these problems:

1.

A. 4.8 ÷ 8 = B. 7.2 ÷ 9 =

C. 2.4 ÷ 6 = D. 3.2 ÷ 4 =

E. 1.8 ÷ 3 =

2.

A. 5.6 ÷ 0.7 = B. 6.3 ÷ 0.7 =

C. 2.7 ÷ 0.3 = D. 4.9 ÷ 0.7 =

E. 2.8 ÷ 0.7 =

3.A. 1.65 ÷ 0.15 = B. 27.3 ÷ 1.3 =

C. 0.03 ÷ 0.005 = D. 0.99 ÷ 0.0009 =



There’s nothing mysterious! An exponent is simply shorthand for

multiplying that number of identical factors. So 4³ is the same as(4)(4)(4), three identical factors of 4.

Multiplying exponents with the same base:

When you are multiplying terms with the same base you can add

the exponents.

This means: 4 x 4 x 4 x 4 x 4 x 4 x 4 or 4 · 4 · 4 · 4 · 4 · 4 · 4

When you are dividing terms with the same base you can subtract

the exponents.

This means: 4 x 4 x 4 or 4 · 4 · 4

Try these:

Exponents

1.

A. 106 = B. 51 =C. 62 = D. 71 =E. 61 = F. 42 =

3.A. 21 x 24 = B. 50 x 53 =

C. 4-2 x 45 = D. 252 x 1253 =



Perfect Squares and

Square RootsWe know that to square a number we just multiply it by itself.

A square root goes the other way:

3 squared is 9, so a square root of 9 is 3

A square root of a number is a value that can be multiplied by

itself to give the original number.

A square root of 9 is 3, because when 3 is multiplied by itself you

get 9.

1 Squared = 12 = 1 × 1 = 1

2 Squared = 22 = 2 × 2 = 4

3 Squared = 32 = 3 × 3 = 9

4 Squared = 42 = 4 × 4 = 16

5 Squared = 52 = 5 × 5 = 25

6 Squared = 62 = 6 × 6 = 36



The Square RootThis is the special symbol that means "square root",

you would say "square root of 9 equals 3"

Example: What is √25?

We know that 25 = 5 × 5, so if you multiply 5 by itself (5 × 5) you

will get 25.

So the answer is: √25 = 5

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The perfect squares are the squares of whole numbers

Try to remember at least the first 10 of those.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 et

c Perfect

Squares: 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ...



Now try this activity:

Join up the numbers to their correct square numbers and complete

the original number (the square root) in the column to the right.

Square sum

2 x 2

5 x 5

6 x 6

12 x 12

9 x 9

1 x 1

3 x 3

8 x 8

7 x 7

11 x 11

4 x 4

10 x 10

Square number Square root

81

9

1

49

25

4

121

16

100

144

36

64



Order of OperationsSome expressions include powers and other operations.

BIDMAS gives the order in which operations should be carried out.

Remember that BIDMAS stands for:

Brackets, Indices, Division and Multiplication , Addition and Sub-

traction

If there are brackets, work out the value of the expression in

the brackets first.

Square roots are carried out at the same stage as indices.

If there are no brackets, do multiplication and division before

addition and subtraction no matter where they come in an ex-

pression.

If an expression has only addition and subtraction then work it

out from left to right.

Try these examples:

= ................................262 =. ....................................

32 1010

= .................................4216 = ....................................2)42(

= . ................................3)5212( = ...................................49



Try this BIDMAS activity

A student hands in this work on evaluating formulas. Using the rules

of BIDMAS identify and correct the mistakes they have made:

1. 22 – 5 x 4 + 10

= 17 x 4 + 10

= 78

2. 30 - 3 x 7 + 5

= 30 – 21 + 5

= 30 – 26

= 4

3. 3 x 4 + 1 x 6

= 13 x 6

= 78

4. 2(4 + 7)

= 2 x 4 + 7

= 15

5. 5(2 + 7) + 1

= 5 x 2 + 5 x 7 + 5 x 1

= 50

6. (2 + 5)2

= 22 + 52

= 29

7. ½ (42 + 6)

= ½ x 42 + ½ x 6

= 22 + 3

= 7

8. 2(1 + 4)2

= 22 x (1 + 4)2

= 4 x 25

= 100



Problem Solving and

Application of Mathematics

in the Real WorldWe have so far looked at example questions with numbers only.

Maths can help us solve real life problems. In this section you will

use your mathematical skills to problem solve some real life

problems:

1. A child is entered into the hospital after ingesting 12 aspirin

tablets. The Merck Index indicates that renal failure can occur if

as little as 3 grams is ingested, and may be fatal if as much as

10grams is eaten. If each aspirin tablet contains 300 mg of aspirin,

is the child in danger of death or renal failure?

Hint: Some of the terms here may be unfamiliar

to you, but look and the numeric values and try

to work out what the question is asking you to

solve.

Write your solution here:



2. A hospital can purchase soap

solution in two ways: 500 mL for

130 dirham’s or 4L for 535

dirham’s. Which product is the

better buy?


3. Estimate the number of years an average person sleeps in alifetime.

Hint Guess how many hours an average person sleeps per night. You

must consider that older people may only sleep a few hours per

night, and infants sleep many more hours per day.




4. Marina Mall has 102 shop outlets, 46 of these are food outlets,

12 are electrical shops, and the rest are fashion shops. If the

rent charged for an outlet is 2030.65 dirham’s per week,

determine the revenue that the mall receives through rent

obtained from fashion shops per month.


RevisionThe following website has revision exercises and task for you

complete. You can use this resource to revise number theory for

your end of module exam

http://www.bbc.co.uk/schools/gcsebitesize/maths/number/

See how much you have learned, try these revision exercises on the

next page.






A. 5 x (1 + 4) B. 7 – (5 + 1)

C. 7 + 2 x 4 – 12 ÷ 3 D. 56 ÷ 8 – 2

E. 14 + 2 x 9

Work out

Work out

A. 33 B.

C. 33+22 D.

Add brackets to make the calculation correct

A. 6 + 14 ÷ 2 = 10 B. 9 + 15 ÷ 8 = 3

C. 8 + 2 x 6 – 5 = 10 D. 51 ÷ 3 + 5 = 22

E. 11 + 6 x 4 = 68 F. 10 – 2 + 22 = 4

G. 3 + 5 x 9 – 8 = 8 H. 8 + 15 ÷ 3 x 0 = 0



A. 0.3 x 0.5 B. 0.09 x 0.08

C. 14 ÷ 0.2 D. 52.05 ÷ 0.15

E. 7.533 ÷ 0.31 F.

Work out

Given that 6.3 x 2.7 = 17.01 work out

A. 63 x 27

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

B. 630 x 2.7

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

C. 0.63 x 27

…………………………………………………………………………………………………………...



Basic Mathematical

NotationSymbol How it is read in English

+ plus, positive

− minus, negative

± plus or minus

× times, multiplied by

÷, / over, divided by

= equals, is equal to

( ) parentheses, brackets

. point

Documents

AFM7000 Student Workbook