Topic 3: Fractions -...

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Topic 3: Fractions

Topic 1

Integers

Duration

1 1/2 weeks

Content Outline

PART 1 (1/2 week)

Introduction

Converting Fractions to Decimals

Converting Decimals to Fractions

Equivalent Fractions

Simplifying Fractions

Mixed, Improper and Proper Fractions

Lowest Common Denominator

PART 2 (1 week)

Addition/Subtraction of Fractions with Common Denominators

Addition/Subtraction of Fractions with Different Denominators

Multiplication of Fractions

Reciprocals

Division of Fractions

Cancelling

Fractions and Chance

Topic 2

Decimals

Topic 3

Fractions

Topic 4

Ratios

Topic 5

Percentages

Topic 6

Algebra

Topic 7

Equations and Formulae

Topic 8

Measurement

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Topic 3: Fractions

Introduction

Fractions are used to describe anything that has been divided into equal sized

pieces. Some examples are a block of chocolate, an apple or a litre of lemonade.

Fractions state how many pieces something is divided into or shared between and

how many are chosen. How does it work?

A fraction represents a part of a whole.

4

1

top number or NUMERATOR

bottom number or DENOMINATOR

The Numerator refers to the number of parts you have.

The Denominator is the number of equal size pieces whole is divided into.

Example 1: Consider the following:

The shaded portion of the pie represents 56 or 5/6 or 5

6 of the total pie; there are 5 pieces chosen out of the 6

pieces the shape is divided into.

The shaded portion of the box represents 312

(or

14

). There are 3 pieces chosen out of the 12

pieces the shape is divided into.

This piece is NOT broken up into equally shaped

pieces. This cannot be described as the fraction 13

.

55

Converting Fractions to Decimals

A fraction can be considered to indicate a division:

numeratornumerator denominator

denominator

To change a fraction to a decimal, divide the

numerator by the denominator.

Example 1: Convert 3

4 to a decimal.

Step 1: Consider the fraction as a division.

434

3

Step 2: Perform the division:

3 2

0.75

4 3. 0 0

3

0.754


3 to a decimal.

Step 1: Consider the fraction as a division.

2

2 33


2 2 2

0.66.....

3 2. 0 0 0..

2 3 0.666..

This is written in short form as 0.6

Some common fractions you may

know and not need to work out.

This is called a

recurring decimal

56


11 to a decimal.

Step 1: Consider the fraction as a division and perform the division.

9 2 9 2

0. 8 1 8 1 ...

11 9. 0 0 0 0..


90.8181.. 0.81

11

Converting Decimals to Fractions

Look at the following place value table:

decimal units

.

tenth

s

hundre

dth

s

thousandth

s

ten

thousandth

s

fraction

0.4 = 0 . 4 10

4

0.04 = 0 . 0 4 100

4

0.004 = 0 . 0 0 4 1000

4

0.0004 = 0 . 0 0 0 4 4

10000

0.25 = 0 . 2 5 100

25

100

5

10

2

0.257 = 0 . 2 5 7 1000

257

1000

7

100

5

10

2

4.368 = 4 . 3 6 8 3 6 8

410 100 1000

=368

41000

If two or more digits repeat

then we place a line above the

group of digits that repeat.

57

To change a decimal to a fraction express the fraction in terms of

its place value.

The number of places after the decimal point tells us how many

zeros to use in the number under the fraction line, i.e. whether it

should be 10, 100, 1000…

Example 1: Convert 0.14 to a fraction.

Step 1: Since there are two digits after the decimal point, the fraction will

have a denominator of 100.

14

0.14100

Step 2: Where possible, simplify the fraction.

2

14 7

100 50

2

Example 1: Convert 0.625 to a fraction.

Step 1: Since there are three digits after the decimal point, the fraction will

have a denominator of 1000.

625

0.6251000

Step 2: Simplify the fraction.

25 5

625 25

1000 40

5

8

25 5

0.14 is called a decimal

fraction and 7/50 is

called a common

fraction

Simplifying fractions is

explained in more detail

to follow.

58

Equivalent Fractions

Equivalent Fractions are equal in value, even though they may look different.

You can imagine it like this: A block of wood is cut up into pieces

2

1 (half) of the wood

(1 out of the 2 pieces).

4

2 of the wood


of the piece of wood


You can see that the fractional amount is the same size no matter how many

pieces it is cut into.

You can see that 2

1 can be cut and shown as

4

2 or

8

4

1 2 4

2 4 8

Why are they the same?

Because when you multiply or divide both the top and bottom by the same

number, the fraction keeps it's value.

The rule to remember is:

What you multiply or divide numerator by

you must also do to the denominator!

2 2

1 2 4

2 4 8

2 2

8

4

These are all equivalent, they

represent the same amount.

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Example 1: Express 6

3 as

2

?

Step 1: Look at the fractions you want to change. It will always be changed by

multiplying or dividing.

To work it out, ask yourself:

“What do I need to do to change the denominator from a 6 to a 2?”

To make something smaller, you need to break it down (divide).

6 3 = 2

Step 2: Whatever you do to the top, you must do to the bottom.

In this example we divide the bottom by 3 (to make it a 2), so we will

also have to divide the top by 3 to find the equivalent fraction.

3 3becomes

1

6 3 2


3 as

12

?

Step 1: To work it out, ask yourself:

“What do I need to do to change the denominator from a 4 to a 12?”

To make something larger, you need to multiply it.

4 x 3 = 12

Step 2: Whatever you do to the top, you must do to the bottom.

In this example we multiply the bottom by 3 (to make it a 12), so we

will also have to multiply the top by 3 to find the equivalent fraction.

3 x 3becomes

9

4 x 3 12

Important to remember:

The top and bottom of the fraction must always be a whole number.

Hence, the number you pick to divide by must always divide evenly (ie

no remainders) for both the top and bottom numbers.

You only multiply or divide, never add or subtract, to get an equivalent

fraction.

60

Simplifying Fractions

Simplifying (or reducing) fractions means making the fraction as simple as

possible. Why say three-sixths ( 36

) when you can say a half ( 12

)?

Sometimes we encounter fractions with extremely large denominators. It is useful

to reduce such fractions to a more manageable size. We can use the principle of

equivalent fractions to do this.

To simplify a fraction, divide the numerator and denominator by the

highest number that can divide into both numbers exactly.

Example 1: Simplify 24

108

Step 1: When simplifying, the fraction will always be changed by dividing

since we want to remove common factors.

To work it out, ask yourself:

“What number goes into both the numerator and the denominator?”

Whatever you do to the top, you must do to the bottom.

Step 2: In this example the numerator and denominator have been divided

by 2, then 2 and then 3 to get the final answer.

÷ 2 ÷ 2 ÷ 3

24 12 6 2

108 54 27 9

÷ 2 ÷ 2 ÷ 3

Step 3: Check: Can this be broken down any further?

“What number goes into both the numerator and the denominator?”

Only 1 goes into 2 and 9 and this will not change the result.

The fraction 24

108 has been reduced to the equivalent fraction

2

9.

You may have been able to do

this in less steps

61

Mixed, Improper and Proper Fractions

Proper fractions have a numerator smaller than the denominator. These are

probably the fractions with which we are most familiar.

For example: 4

3,

8

5 and

10

1

Improper fractions have a numerator larger than the denominator (they are ‘top

heavy’).

For example: 2

7,

3

5 and

5

14

Mixed numbers contain both a whole number part and a fraction part.

For example: 3

21 and

4

35

Example 1: If we have 3 and ½ buckets of water, we are using mixed numbers.

Each whole is made up of 2 halves, so 3 wholes make 6 halves. And then there is

another half.

So 3 and ½ buckets is the same amount as 7 halves

2

7

2

13

We need to be able to make the conversion between mixed and improper fractions

without drawing pictures.

To convert a mixed fraction to an improper fraction:

Multiply the whole number part by the fraction's denominator.

Add that to the numerator

Write the result on top of the denominator.

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34 as an improper fraction.

Step 1: The answer will be in fifths (since the denominator is 5).

First we need to calculate how many fifths are in 4 wholes.

4 x 5 = 20

Step 2: We add the 3 fifths.

20 + 3 = 23

Step 3: 5

34 = (4 × 5 + 3) fifths = 23 fifths =

5

23

To convert an improper fraction to a mixed fraction:

Divide the numerator by the denominator.

Write down the whole number answer

Then write down any remainder above the denominator.


59 as a mixed number.

Step 1: The question asks you to change 59 quarters into a whole number

part and a fraction part.

The answer will be in quarters (since the denominator is 4).

Step 2: First we need to calculate how many times 4 goes into 59.

459 = ??

Hint You may need to do long division.

1

14

4 5 9

Step 3: This means that 4 goes into 59 14 times with a remainder of 3, so

4

59=

4

314

remainder 3 remainder 3

63

Lowest Common Denominator

Finding a lowest common denominator allows you to compare the same type of

fractions, ones that have the same size pieces. It is useful to be able to find the

lowest common denominator of fractions.

Once again equivalent fractions will be used.


3and

4

1as equivalent fractions with the lowest

common denominator.

Step 1: Using equivalent fractions we can see that

24

9

16

6

8

3 ….. and

20

5

16

4

12

3

8

2

4

1 …..

Look at the denominators of the fractions. Is there one that is the same?

8 (eighths) and 16 (sixteenths) appear in both, but 8 is smaller (lower).

Step 2: 8 is the lowest common denominator of 8

3and

4

1.

So the answer is 8

3and

8

2

SHORT CUT:

Step 1: You don’t need to calculate all the numerators when asked to find the

LCD, just look at the denominators.

8

3will have equivalent fractions with denominators of

8, 16, 24, 32… (8 times table)

4

1 will have equivalent fractions with denominators of

4, 8, 12, 16, 20…(4 times table)

The lowest common denominator will be 8.

Step 2: Express both 8

3and

4

1 with a denominator of 8.

8

3does not need to be changed

1 X 2becomes

2

4 X 2 8

Step 3: So the answer is 8

3and

8

2

64

Example 3: Which is the larger fraction: 3

4 or

2

3 ?

Step 1: Find the lowest common denominator (LCD) of the two fractions.

3

4 can have equivalent fractions with denominators of 4, 8, 12, 16, 20…

2

3 can have equivalent fractions with denominators of 3, 6, 9, 12, 15 …

The LCD of quarters and thirds is 12.

Step 2: Express both fractions with the common denominator of 12:

3 3 3 9

4 4 3 12

,

2 2 4 8

3 3 4 12

Step 3: It is now obvious which is the larger of the two fractions:

9 8

12 12 means that

3 2

4 3

When you wish to add, subtract, or determine the larger of two or more fractions,

you will use the principle of equivalent fractions with lowest common

denominators.

mathematical symbol meaning

‘greater than’

65

Addition and Subtraction of Fractions with Common Denominators

There are three types of questions involving adding or subtracting fractions: easy,

not so easy and messy. The difference is with the denominators.

When fractions have the same denominators it means the individual pieces are of

equal value. When you add them, you are counting how many quarters of an

orange you have, squares of chocolate in each child’s hand or sandwich halves.

How do we turn this into the rules we need for maths?

Example 1: Imagine an orange cut into quarters.

There are two pieces left.

4

2

4

1

4

1

Example 2: A chocolate bar is cut into 12 pieces. You want 5 pieces and

your friend wants 2 pieces.

Step 1: Count up the pieces you want: 2 + 5 = 7

Step 2: The piece sizes do not change, so the denominator stays as twelfths.

12

7

12

25

12

2

12

5

To add or subtract fractions with common denominators

Add the numerators and leave the denominator the same.

Give your answer as a simplified, mixed number (if needed).

Say the question out loud:

one quarter plus one quarter gives 2

quarters.

Say the question out loud:

Five twelfths plus two twelfths gives 7 twelfths.

Improper fractions are converted to mixed numbers. If the

answer to a question was 92

, it is more commonly called 4 12

.

66

Addition and Subtraction of Fractions with Different Denominators

There is a bit of work to do before the answer can be calculated.

To add or subtract fractions with different denominators

find a common denominator by using equivalent fractions

then follow the rule of similar denominators above.

Let’s try out an example that we can imagine easily (a good way to work out any

problem).

Example 1: Calculate4

1

2

1

Step 1: Problem!! The question involves halves and quarters.

Solution: Find a common denominator.

The denominators for 2

1 would be 2, 4, 6, 8…

The denominators for 4

1 would be 4, 8, 12, 16…

The common denominator of a half (2) and a quarter (4) is 4.

Step 2: Using equivalent fractions:

1 X 2becomes

2

2 X 2 4

Step 3: Change both the fractions to quarters

4

1

2

1

4

1

4

2

4

12

4

3

Now that the denominators are

the same, you can add the

numerators, leaving the

denominator the same.

67

Example 2: Calculate4

1

5

2

Step 1: Find a common denominator.

The denominators for fifths (5) would be 5, 10, 15, 20, 25…

The denominators for quarters (4) would be 4, 8, 12, 16, 20,

24…

The common denominator for 5 and 4 would be 20.

Step 2: Using equivalent fractions

2 X 4=

8 1 X 5=

5

5 X 4 20 4 X 5 20

Step 3: Change the denominator to 20

4

1

5

2

20

5

20

8

20

58

20

3

Example 3: 3

12

2

11

Step 1: Put integers and fractions together.

3

12

2

11 = )

3

1

2

1()21(

Step 2: Add integers and fractions separately.

)3

1

2

1()21( =

6

2

6

33

= 6

53

Now that the denominators are the

same, you can subtract the

numerators.

WHEN ADDING: If integers are

involved, add the integers, then add

the fractions.

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Example 4: 2 2 17 5

3 15 3 5 3

51 25

15 15

26

15

11

115

Multiplication of Fractions

Compared to adding and subtracting, multiplying fractions is a breeze!

To multiply fractions:

Change mixed numbers into improper fractions

Multiply the numerators together

Multiply the denominators together

Give your answer as a simplified, mixed number

Example 1: What is half of a half?

Answer: If you think about halving half an apple, you get the

answer of one quarter

If you were to write the above as a mathematical statement, you

would have:

4

1

2

1

2

1

‘of’ means ' ’

WHEN SUBTRACTING: If

integers are involved, it is

better to change the mixed

number to an improper

fraction before carrying out

the subtraction:

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Example 2: What is one third of a quarter?

Answer: Look at the diagram below, where a quarter has been divided into

three parts

... and the answer is a twelfth, 1

12

If you were to write the above as a mathematical statement, you would have:

1 1 1

3 4 12

Example 3: Calculate 5

2

4

3

Step 1: Multiply the numerators and

denominators together.

Step 2: Simplify the fraction

54

23

5

2

4

3

6

3

1020

10

3

NOTE: In the example above you can remove common factors before

multiplying. If you do this you don’t need to simplify the fraction and

the multiplication will be easier.

2

3 2 3

4 5 4

12

5

= 3 1 3

2 5 10

One quarter has been divided

into 3 equal parts.

What number goes into both 6 and 20?

Answer: 2 does

In this case,

divide the numerator and denominator by 2.

70


12

4

31

Step 1: Change to improper fractions. 3

7

4

7

3

12

4

31

Step 2: Multiply the numerators and

denominators together.

Step 3: Simplify the fraction.

34

77

12

49

Step 4: Give answer as mixed number

12

14


55

6

55 =

6

5

1

5

= 61

55

= 6

25

= 6

14

What number

goes into

both 49 and

12?

Answer: none,

it cannot be

simplified.

Whole numbers are best

written in fraction form.


You do not need common denominators to multiply fractions

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Reciprocals

The reciprocal of a fraction can be found by inverting or ‘flipping’ the fraction. It

is used in division of fractions.

Proper fractions: the reciprocal of 4

3 is

3

4

Mixed numbers: the reciprocal of 2

14 is found by changing the mixed number to

an improper fraction

2

9

2

14 so the reciprocal is

9

2

Whole numbers: the reciprocal of 5 is found by writing 5 as 1

5

1

55 so the reciprocal of 5 is

5

1

Division of Fractions

To divide fractions, we use the fact that division is the inverse process to

multiplication.

To divide fractions:

Change mixed numbers into improper fractions

Invert (flip) the number after the division sign

Change the to x

Multiply as before

Consider the following examples:

Example 1: How many halves in 7?

Think about it, there are 2 halves in every whole. So there must be 14 halves in 7.

The answer is bigger than the question!

Step 1: Write the question as a mathematical statement 2

17

Step 2: Multiply by the reciprocal 1

2

1

7

= 14

72

Example 2: If I have $10 and divide it into 5 equal shares, how much is each

share?

Answer: 10 5 2

If I take a 1

5 share of $10, how much do I have?

Answer: 1

5of 10 =

110

5 = 2


2

7

6

Step 1: Turn this into a multiplication problem.

change the division into a multiplication sign.

‘flip’ the fraction that follows the division sign.

5

2

7

6

2

5

7

6

Step 2: Simplify and give answer as a mixed number.

27

56

1530

14 7

1

27


1218

Step 1: Change each number to an improper fraction. 4

9

1

18

4

1218

Step 2: Turn into a multiplication problem. 9

4

1

18

91

418

9

72

Step 3: Simplify.

What number goes into both 72 and 9?

Answer: 9 does.

72

8

9 1

= 8

73

Cancelling

When simplifying fractions, we divided both numerator and denominator by

common factors. This technique can be used to advantage when multiplying

fractions.

This process is sometimes called cancelling, and should be continued until no

further common factors remain between numerators and denominators.

Examples:

a) 3

2

5

3 =

35

23

1

1

= 15

21

= 5

2

b) 12

5

6

5 =

5

12

6

5

=

21

5 12

1 156

= 11

21

= 2

c) 29

61

8

53

5

13 =

29

35

8

29

5

16

=

2

16 291

357

115 8 29

1

= 111

712

= 14


Cancelling cannot be used when additions or subtractions appear in numerator

or denominator.

74

Order of Operations and Fractions

The rules we used in Topic 1 for order of operations apply to fraction calculations.

Example 1: 1 1

3 1 44 2

Step 1:

1 13 1 4

4 2

= 5 9

34 2

= 5 18

34 4

= 23

34

…using BEDMAS calculate the brackets

first

… need to get a common denominator when

adding fractions

Step 2: =

3 23

1 4

= 69

4 =

117

4

…multiplication is next

Fractions and Chance

Card games, horse races, rolling dice, insurance and weather forecasts are some of

the everyday occurrences which rely on describing in numbers the chance of an

event happening. Fractions are ideal for this purpose.

Example 1: Here is a hand of playing cards. If you ask a friend to select

one card at random (without looking) what would be the

chance of selecting a King?

Answer:

Your answer should be ‘one out of 5’,

which is easily written as the fraction 15

75

Example 2: Here is another hand of cards. If you selected one at random,

what would be the chance of selecting a 2? Also, what would

be the chance of selecting an Ace?

Chance of an event = Number of times the event occurs

Number of possible outcomes

In the last example, the sample space is A, A, A, 2, 2 so the number of possible

outcomes is 5. Since 3 of these 5 outcomes were Aces, we write:

Chance of Ace = 5

3

Example 3: A coin is thrown 10 times and the following is recorded where

H is heads and T is tails.

H T T H H T H T T T

In this experiment, what is the chance of having a tail appear?

Chance of an event = Number of times the event occurs

Number of possible outcomes

The number of times the event occurred was 6, and the sample space was 10, so

Chance of throwing a Tail = 10

6 =

3

5

A list of all possible outcomes is

called a sample space.

You probably had little trouble

deciding that the chance of selecting a

‘2’ was 2 out of 5 or 5

2 and the chance

of selecting an Ace was 3 out of 5 or 5

3 .

Topic 3: Fractions -...

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