What is a fraction?

Cut a cake into 4 equal pieces and eat 3 of them, and you have eaten three quarters of the cake. A fraction is the way of writing that down: 3/4. It needs two numbers, one for the size of the pieces and one for how many of them you have.

The rule

A fraction is part of a whole that has been cut into equal parts. The bottom number, the denominator, says how many equal parts the whole is cut into. The top number, the numerator, says how many of those parts you have.

How to read a fraction

Count how many equal parts the whole is cut into. That is the bottom number.
Check that the parts really are the same size. If they are not, you cannot name the fraction by counting them.
Count the parts that are shaded, or eaten, or taken. That is the top number.
Say it as "top number, then the name of the part": 3/4 is three quarters, 2/5 is two fifths and 5/8 is five eighths. 1/2 is a half and 1/3 is a third.
To find a fraction of a number of things, divide by the bottom number to get one part, then multiply by the top number to get the parts you want.

Worked examples

What fraction of the bar is shaded?

The bar is cut into 4 equal parts, so the bottom number is 4. 3 of them are shaded, so the top number is 3. The shaded part is 3/4, three quarters.

What fraction of the pizza is shaded?

The pizza is cut into 8 equal slices and 3 are shaded, so 3/8 of it is shaded: three eighths. The 5 slices left are 5/8, and 3/8 and 5/8 make the whole pizza.

What is 3/4 of 12 sweets?

12 ÷ 4 = 3 sweets in each quarter
3 × 3 = 9 sweets in three quarters

Share the 12 sweets into 4 equal groups, one for each quarter. Each group has 3. Three of the groups hold 9 sweets, so 3/4 of 12 is 9.

Is the shaded part a quarter of the bar?

No. The bar is in 4 parts, but they are not equal, so the shaded part is not a quarter. It is as long as the other three put together, so it is really a half.

Unit fractions

A fraction with 1 on top is called a unit fraction: 1/2, 1/3, 1/4, 1/10. It is one single part, and every other fraction is made of them. 3/4 is three lots of 1/4.

The bigger the bottom number, the smaller the part, because the whole is shared into more pieces. 1/10 of a cake is much less than 1/2 of it. Comparing fractions builds on this.

When the top and the bottom are the same

4/4 means the whole has been cut into 4 parts and you have all 4 of them. That is the whole thing, so 4/4 = 1. In the same way 2/2, 3/3 and 8/8 are all one whole.

If the top is bigger than the bottom, you have more than one whole. 5/4 is one whole cake and one more quarter, which is also written 1 1/4. 5/4 is called an improper fraction and 1 1/4 a mixed number.

Quarters and fourths

British schools say quarters and American schools often say fourths. Three quarters and three fourths are the same fraction, 3/4. Every other part has one name, like thirds, fifths and eighths.

Why the bottom number names the parts

Read 3/4 as "3 quarters" and it works like "3 apples". The 3 counts, and the word says what is being counted. The bottom number does the job of the word: it says what size the parts are, by saying how many of them fill the whole.

That is why the parts must be equal. "A quarter" means one of 4 equal parts. If the parts are different sizes, "1 part out of 4" does not tell you how much you have, as the bar in the last example shows.

It is also why a fraction of a set works by dividing. 3/4 of 12 sweets means: cut the 12 into 4 equal groups, then take 3 of the groups. Dividing by 4 makes the groups, and multiplying by 3 takes the ones you want.

Where mistakes happen

Writing shaded over unshaded. With 3 parts shaded and 1 not, the fraction is 3/4, not 3/1. The bottom number counts all the parts, shaded or not.
Counting parts that are not equal. A shape cut into 4 uneven pieces does not show quarters, so shading 1 piece does not show 1/4.
Thinking a bigger bottom number means a bigger fraction. 1/8 is smaller than 1/4, because a pizza cut into 8 has smaller slices than one cut into 4.
Dividing by the top number in a fraction of a set. For 3/4 of 12, divide by the 4, the number of parts, to get 3 in each part. Dividing by the 3 gives 4, which is a third of 12, not a quarter.

Questions people ask

Which number is the numerator and which is the denominator?

The numerator is on top and the denominator is underneath. One way to remember: the denominator is down below, and both start with d. The line between them is called the fraction bar.

What does 0/4 mean?

The whole is cut into 4 parts and you have none of them. 0/4 is zero. But 4/0 has no meaning: you cannot cut a whole into 0 parts.

Is a half always 1/2?

A half can be written in many ways. 2/4, 3/6 and 5/10 are all the same amount as 1/2. They are called equivalent fractions.

Do the parts have to be the same shape?

No, only the same size. A square cut corner to corner into 2 triangles shows halves, and so does one cut down the middle into 2 rectangles.