Fractions - Maths & Stats - Basics - The Library at Leeds Beckett University

Learning Outcomes

If you work through this section you should be able to:

Recognise the written format of a fraction and define what a fraction is.
Recognise proper and improper fractions, numerators and denominators.
Add, subtract, multiply and divide fractions.
Simplify fractions.
Convert mixed numbers into fractions and improper fractions into mixed numbers.
Convert a fraction into a decimal.
Convert a fraction into a percentage.

Fractions are simply 'parts of things'. When a whole is divided into parts, these parts are fractions of the whole.

Image of a circle divided into four equal parts. Each part represents a quarter (1/4) of the whole. Three of the parts are shaded the same colour and this represents three quarters (3/4) An example showing parts as fractions of the whole

Fractions are written as two numbers with a line between them.

For example: 2/3, $\frac{3}{5}$ and $\frac{1}{13}$

The number above the line (or to the left) is the numerator.

The number below the line (or to the right) is the denominator.

The denominator states how many small, equal parts the whole unit is divided into; the numerator states how many of the smaller parts are being counted.

An example illustrating the numerator and the denominator

Mixed numbers

Where whole numbers and fractions are written together, these are called mixed numbers.

For example:

$7 \frac{3}{5}$ (Seven and three fifths)

$2 \frac{1}{2}$ (Two and one half)

$1$ $\frac{2}{3}$ (One and two thirds)

If the numerator of a fraction is less than its denominator it is called a proper fraction.

Examples of proper fractions

2/3
$\frac{5}{8}$
$\frac{3}{4}$

If the numerator is greater than its denominator it is an improper fraction. This is sometimes called a top heavy fraction.

Examples of improper fractions

5/4
$\frac{3}{2}$
$\frac{9}{7}$

Improper fractions can also be expressed as mixed numbers (see the 'Introduction' to this section).

You will need to convert improper fractions into mixed numbers and mixed numbers into fractions when doing calculations.

Mixed numbers into fractions

This is finding out how many parts (with the same denominator as the fraction part) there are. The denominator, then, will stay the same but we need to find a new numerator.

Multiply the whole number by the denominator to find out how many parts the whole number represents
Then add the numerator to this.

New numerator = (whole number × denominator) + numerator

Example

Convert $3 \frac{4}{7}$ into a fraction.

Convert the whole part of the number.

We need to convert 3 into $\frac{1}{7}$ parts

(3 × 7) = 21
Add the numerator

21 + 4 = 25

Answer: $\frac{25}{7}$

Improper fractions into mixed numbers

This is finding out how many 'wholes' there are and then how many fractions are left over.

Divide the denominator into the numerator.
The remainder becomes the new numerator. The denominator stays the same.

Mixed number = numerator ÷ denominator

The answer is the whole number and the remainder is the numerator of the proper fraction.

Example

Convert $\frac{9}{4}$ into a mixed number

Divide the denominator into the numerator.

9 ÷ 4 = 2 remainder 1
The remainder becomes the new numerator. The denominator stays the same.

Answer: $2 \frac{1}{4}$

Equivalent fractions are fractions that have the same value but are represented with different denominators.

A rectangle split into five rows. The bottom row is split into sixteen equal parts, the next one up, into eight equal parts, the next, four equal parts, the next, two equal parts, and the top one is whole. The left half of the rectangle is shaded, demonstrating that 1/2, 2/4, 4/8, and 8/16 are equal, or equivalent. Fraction wall illustrating equivalent fractions.

A rectangle divided into four rows. The bottom row is in twenty equal parts, the next one up, in ten equal parts, the next one, in five equal parts, and the top one is whole. From the left, the area up to 12/20, 6/10, and 3/5, is shaded to demonstrate that they are equal, or equivalent. Fraction wall illustrating equivalent fractions.

Simplifying fractions (or cancelling down)

When doing mathematical work it is sometimes more appropriate to work with fractions in their simplest form, i.e. with their smallest possible denominator, as this is more meaningful. So rather than $\frac{12}{20}$ (see example above) we would use $\frac{3}{5}$ which is its simplest form.

Simplifying fractions is sometimes called cancelling down.

What you need to do is:

Find a common factor: a number that will divide into both the numerator and the denominator. There may be more than one but the larger the better.
Divide the numerator and the denominator by the common factor.
Check that there isn't another common factor in your answer. If there is, repeat.

Note: cancelling down can only be carried out between numerators and their denominators.

Example

Simplify $\frac{9}{12}$

Find a common factor.

3 divides into both 9 and 12. So 3 is the common factor.
Divide the numerator and the denominator by the common factor.

9 ÷ 3 = 3

12 ÷ 3 = 4

Answer: $\frac{3}{4}$

There will be some fractions that have more than one common factor.

Example

Simplify $\frac{30}{42}$

30/42 simplifies to 15/21

Observe that 2 is a common factor.
Divide numerator and denominator by 2 (cancel by 2).

15/21 simplifies to 5/7

Observe that 3 is a common factor.
Divide numerator and denominator by 3 (cancel by 3).

Answer: $\frac{5}{7}$

You may have noticed straight away that 6 is a common factor, and, if so, you would be able to do this in one step. It doesn't really matter how many times you repeat the process of cancelling as long as you check that the final fraction has no common factors.

Expressing fractions with their common denominators

Expressing fractions with their lowest common denominator is essential for adding and subtracting fractions. A common denominator is a number that all the denominators will divide into.

First find the lowest common denominator of the set of fractions you are working with. This will be the new denominator for each of the fractions.
For each fraction, divide the original denominator into the new denominator and multiply by the numerator to get the new numerator.

New numerator = (common denominator ÷ original denominator) × original numerator

Example

Express the following fractions with a common denominator: $\frac{1}{3}$ , $\frac{5}{6}$ , $\frac{8}{15}$

Find a common denominator.

The lowest common denominator is 30.
Divide original denominator into new denominator and multiply by numerator.

(30 ÷ 3) × 1 = 10

(30 ÷ 6) × 5 = 25

(30 ÷ 15) × 8 = 16

Answer: $\frac{10}{30}$ , $\frac{25}{30}$ , $\frac{16}{30}$

Fractions can be added and subtracted but first they must have the same denominators. Those with different denominators need to be expressed with a common denominator (see 'Equivalent fractions' to see how to express groups of fractions with their common denominator).

Once your fractions have the same denominator simply add the numerators together.

Example

$\frac{2}{11}$ + $\frac{3}{19}$

209 is the lowest common denominator.

(209 ÷ 11) × 2 = 38

(209 ÷ 19) × 3 = 33

$\frac{38}{209} + \frac{33}{209} = \frac{71}{209}$

You can write it out like this:

$\frac{2}{11} + \frac{3}{19} = \frac{(38 + 33)}{209} = \frac{71}{209}$

If you are left with any improper fractions, convert these into mixed numbers (see 'Converting improper fractions and mixed numbers').
You may also find that you get an answer that is not cancelled down to its simplest form. You should cancel until it is (see 'Equivalent fractions').
If you have to add mixed numbers, add the whole numbers first, then add the fractions as above.

Example

2 $\frac{1}{2}$ + 4 $\frac{5}{6}$

= 2 + 4 + $\frac{1}{2}$ + $\frac{5}{6}$

= 6 + $\frac{3}{6}$ + $\frac{5}{6}$

= 6 + $\frac{8}{6}$ (this fraction needs to be cancelled down to its lowest form)

= 6 + $\frac{4}{3}$ (improper fraction must be converted into a mixed number)

= 6 + 1 $\frac{1}{3}$

= 7 $\frac{1}{3}$

As with addition, to subtract fractions each fraction should have the same denominator. Those with different denominators need to be expressed with a common denominator (see 'Equivalent fractions').

Once the denominators are the same, subtract the numerators.

Example

$\frac{5}{8}$ − $\frac{2}{5}$

40 is the lowest common denominator.

(40 ÷ 8) × 5 = $\frac{25}{40}$

(40 ÷ 5) × 2 = $\frac{16}{40}$

$\frac{25}{40}$ − $\frac{16}{40}$ = $\frac{9}{40}$

You can write it out like this:

$\frac{5}{8}$ − $\frac{2}{5}$ = $\frac{25 - 16}{40}$ = $\frac{9}{40}$

If mixed numbers are involved, the simplest thing to do is to convert all mixed numbers to improper fractions and then carry out the subtraction as above.

If the answer is still top-heavy, convert back to a mixed number.

Example

$4 \frac{1}{6}$ − $2 \frac{3}{4}$

$\frac{25}{6}$ − $\frac{11}{4}$

$\frac{50}{12}$ − $\frac{33}{12}$ = $\frac{17}{12}$

$\frac{17}{12}$ = $1 \frac{5}{12}$

You can write it out like this:

$4 \frac{1}{6} - 2 \frac{3}{4} = \frac{25}{6} - \frac{11}{4} = \frac{50 - 33}{12} = 1 \frac{5}{12}$

Remember: Check that fractions in your answer are in their simplest form. If not you must cancel them down.

To multiply fractions simply multiply all the numerators together and all the denominators together.

Where mixed numbers are involved, they need to be converted into fractions before they can be multiplied. Remember to convert improper fractions in the answer, if any, to mixed numbers (see 'Converting improper fractions and mixed numbers').

Example

$\frac{2}{5}$ × $5 \frac{2}{3}$

= $\frac{2}{5}$ × $\frac{17}{3}$

2 × 17 = 34

5 × 3 = 15

= $\frac{34}{15}$

= $2 \frac{4}{15}$

You can write it out like this:

$\frac{2}{5} \times 5 \frac{2}{3} = \frac{2}{5} \times \frac{17}{3} = \frac{34}{15} = 2 \frac{4}{15}$

Remember: Check that fractions in your answer are in their simplest form. If not, you must cancel them down.

Invert the second fraction (turn it upside down), then multiply the numerators together and the denominators together.

Example

$\frac{3}{5}$ ÷ $\frac{2}{7}$

= $\frac{3}{5}$ × $\frac{7}{2}$

3 × 7 = 21

5 × 2 = 10

= $\frac{21}{10}$

= $2 \frac{1}{10}$

You can write it out like this:

$\frac{3}{5} \div \frac{2}{7} = \frac{3 \times 7}{5 \times 2} = \frac{21}{10} = 2 \frac{1}{10}$

Any mixed numbers need to be converted into fractions before this process. Remember to convert improper fractions in your answer to mixed numbers, as in the example above.

Remember: Check that fractions in your answer are in their simplest form. If not, you must cancel them down.

To convert a fraction to a decimal, divide the denominator into the numerator.

Numerator ÷ Denominator

19/20 = 19 ÷ 20 = 0.95; 1/5 = 1 ÷ 5 = 0.2; 13/19 = 13 ÷ 19 = 0.68 Examples showing how to convert a fraction to a decimal

To find out how to convert decimals to fractions, see the Decimals section.

To convert a fraction to a percentage there are two methods.

Method 1

First convert it into a decimal (numerator ÷ denominator)
Multiply by 100, which simply means moving the decimal point two places to the right.

(Numerator ÷ Denominator) × 100

3/4 = (3 ÷ 4) × 100 = 75%; 2/5 = (2 ÷ 5) × 100 = 40%; 21/23 = (21 ÷ 23) × 100 = 91.3% Examples showing Method 1

Method 2

Multiply the fraction by 100/1
Cancel as much as possible

3/4 × 100/1 = 3/1 × 25/1 = 75%; 13/30 × 100/1 = 13/3 × 10/1 = 130/3 = 43 1/3 % Examples showing Method 2

To find out how to convert percentages to fractions, see the 'Percentages and ratios' section.

Activity

You can download a version of this Fractions activity in Word format:

Fractions activity

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Skills for Learning: Maths & Stats - Basics

Learning Outcomes

Mixed numbers

Examples of proper fractions

Examples of improper fractions

Mixed numbers into fractions

Example

Improper fractions into mixed numbers

Example

Simplifying fractions (or cancelling down)

Example

Example

Expressing fractions with their common denominators

Example

Example

Example

Example

Example

Example

Example

Method 1

Method 2

Activity

Links

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