Fractions Explained

A fraction describes part of a whole that has been split into equal parts. If a pizza is cut into 8 equal slices and you eat 3, you have eaten 3/8 of it. The word 'equal' matters: if the slices are different sizes, counting them does not tell you what fraction you have.

A fraction is also a number in its own right, and it is a division. 3/4 means 3 ÷ 4, which is 0.75. That is why fractions can be placed on a number line, compared, added and multiplied just like whole numbers.

Every fraction has two parts, separated by a line:

• The denominator is the bottom number. It says how many equal parts the whole is split into, so it names the size of each part. In 3/8 the parts are eighths.
• The numerator is the top number. It counts how many of those parts you have. In 3/8 you have 3 of them.

A quick way to remember which is which: the denominator is down. The denominator can never be 0, because a whole cannot be split into zero parts.

A unit fraction has a numerator of 1: 1/2, 1/3, 1/4, 1/10 and so on. It is one single part of the whole. Every other fraction can be built from unit fractions: 3/8 is 1/8 + 1/8 + 1/8.

With unit fractions, a bigger denominator means a smaller fraction. Share a cake between 8 people instead of 4 and each slice gets smaller, so 1/8 is less than 1/4. This surprises many people at first, because 8 is bigger than 4.

• A proper fraction has a numerator smaller than its denominator, such as 3/5 or 7/10. It is worth less than 1.
• An improper fraction has a numerator bigger than its denominator, such as 9/4. It is worth more than 1.
• A mixed number is a whole number and a proper fraction together, such as 2 1/4 (two and a quarter).

Improper fractions and mixed numbers are two ways of writing the same amount.

Improper to mixed: divide the numerator by the denominator. For 9/4, 9 ÷ 4 = 2 remainder 1, so 9/4 = 2 1/4.

Mixed to improper: multiply the whole number by the denominator and add the numerator. For 2 1/4, 2 × 4 + 1 = 9, so 2 1/4 = 9/4.

Mixed numbers are easier to picture in everyday life (two and a quarter pizzas). Improper fractions are usually easier to calculate with, especially when multiplying and dividing.

To place 3/4 on a number line, split the gap between 0 and 1 into 4 equal steps. Each step is 1/4. Count 3 steps from 0 and you land on 3/4. Keep counting past 1 and you reach 5/4, which is the same point as 1 1/4.

The number line shows two useful facts. Proper fractions always lie between 0 and 1, while improper fractions lie beyond 1. And different fractions can land on exactly the same point: 1/2 and 2/4 sit in the same place because they are equal. The next part of this series explains these equivalent fractions.

Two simple cases need no working out:

• Same denominator: the parts are the same size, so the bigger numerator wins. 5/9 is more than 4/9.
• Same numerator: you have the same number of parts, so the smaller parts lose. 3/10 is less than 3/7, because tenths are smaller than sevenths.

When both numbers are different, rewrite the fractions with a common denominator first, or turn them into decimals.