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Fractions

Equivalent Fractions

This rectangle has been divided into eight equal parts.
Each part is one eighth of the rectangle ().

Shading fractions of diagrams

We can shade three quarters of the rectangle as below.

1
4

Note that each quarter is made up of two eighths. So is the same as .

These are called equivalent fractions. We say = .

If we multiply the top and the bottom of a fraction by the same number, we get an equivalent fraction.

= (x top and bottom by 3).

= (x top and bottom by 4).

Other Types of Fractions

1. Improper or top-heavy fractions:
Where the top number is bigger than the bottom. For example, is a top-heavy fraction.

2. Mixed numbers:
This is a mixture of whole numbers and fractions. For example, .

This can be changed to a top-heavy fraction as follows:

2 x 1+1
2

We multiply the whole number by the bottom number of the fraction and add the top of the fraction, i.e. 2 x 1 + 1 = 3, over the bottom, which gives

e.g.

2
3

3 x 5 + 2
3

17
3

Changing a top-heavy to a mixed fraction:

17
3

= 17 ÷ 3 = 5, remainder 2

This is written as 5 and

2
3

Cancelling a Fraction

This involves converting a fraction to its simplest form (or lowest terms).

We do this by dividing the top and bottom by the same number.

Example:	= (÷ top and bottom by 5) = (÷ 2) = (÷ 4)
Note:	, and cannot be simplified further.

Writing as a fraction

Example: I have 20 sweets and I eat 15. What fraction have I eaten?

The fraction eaten is 15 out of 20. This is written as a fraction, i.e. .

We must always simplify so = (÷ 5).

Finding ‘a fraction of'

Example 1: What is of 8?

Rule: Divide by the bottom and multiply (times) by the top.

8 ÷ 4 x 3 = 6

So of 8 = 6

Example 2:

of 15 = 15 ÷ 5 x 2 = 6

Adding and Subtracting

Examples:

–

When the bottom numbers are the same, we add or subtract the top numbers.

When the bottom numbers are different, we must make them the same by multiplying.

Example 1:

= ?

2 x 3 = 6

By making the bottom numbers 6, we must multiply the top numbers by the same amount.

In other words, = and = and these can now be added to get .

Example 2:

– = ?

(5 x 3 =15)

= and =

So, – =

Mixed numbers

We can add or subtract the whole numbers and then the fractions.

Example:

–

Multiplying

When multiplying fractions, we multiply the top numbers and multiply the bottom numbers. Then we simplify the answer, if possible.

Example 1:

3 x 2
4 x 3

If the numbers are large, we can cancel diagonally.

Example 2: x = ?

15 and 35 cancel by 5 giving x .

16 and 24 cancel by 8 giving and .

So x = x = .

For mixed numbers

Example 3:

= ?

Change to a top-heavy fraction, so x

= 7

Dividing

Rule: Change the sign from ÷ to x and turn over the second fraction. Then use the same method as multiplying.

Example:

÷ = x = = 1

For mixed numbers

Change to a top-heavy fraction first, then use the above rule.

Changing a Fraction to a Decimal

Rule: Divide the top by the bottom.

Example 1:	= 3 ÷ 4 = 0, 75

Or	= = 0,75	(x top and bottom by 25)


Example 2 :	= 3 ÷ 5 = 0,6

Or	= = 0,6	(x top and bottom by 2)