What a fraction is

In 3/8, the 8 says how many equal parts the whole was cut into, and the 3 says how many of those parts you have.

Equal parts of one whole

Cut a pizza into 8 equal slices and eat 3 of them, and you have eaten 3/8 of the pizza. That short expression carries two separate pieces of information. The bottom number, 8, tells you how many equal parts the whole was cut into. The top number, 3, tells you how many of those parts you are talking about. A fraction is a way of writing a part of something, or a number that sits between whole numbers, using those two counts.

The bottom number is called the denominator and the top number is the numerator. One way to keep them apart: the denominator names the kind of piece (eighths, thirds, tenths) the way a noun names a thing, and the numerator counts how many of those pieces you have. The line between them is the fraction bar. When a fraction is written on one line, as in 3/8, the slash does the same job.

The word equal matters more than anything else in the definition. If you cut a cake into 4 pieces of different sizes and take one, you do not have 1/4 of the cake. You have one piece, and nobody can say how big it is without measuring it. A denominator only tells you the size of a part when all the parts are the same size.

Use the bar above to build fractions. Choose how many equal parts the bar has, then click parts to shade them. The line under the bar shows the fraction, the same amount as a decimal, and as a percent. Shade 2 parts out of 8, then switch to 4 parts and shade 1: the shaded length is the same, and both read 0.25.

Unit fractions are the building blocks

A fraction with numerator 1, such as 1/2, 1/5 or 1/100, is a unit fraction. It is one single piece of the whole. Every other fraction can be built by putting unit fractions together: 3/8 is three pieces of size 1/8, or 1/8 + 1/8 + 1/8. This picture of a fraction, a count of equal pieces, is the most useful one to carry around, and later units lean on it again and again. In English, fractions are read with the counting number first and the size of the piece second: one third, three quarters, five eighths.

Fractions bigger than one

Nothing stops the numerator from being bigger than the denominator. If you have 7 quarter cups of flour, you have 7/4 cups, which is more than one full cup. A fraction like 7/4, whose numerator is greater than or equal to its denominator, is called an improper fraction. The name is old and a little misleading, because there is nothing wrong with it; improper fractions are often the easiest form to calculate with. A fraction whose numerator is smaller than its denominator, such as 3/4, is a proper fraction and is always less than 1.

Two special cases are worth knowing. When the numerator equals the denominator, as in 4/4 or 9/9, you have all the parts, so the fraction equals 1. When the denominator is 1, as in 5/1, the whole was not cut at all, so the fraction is the whole number 5. A denominator of 0 is not allowed: you cannot cut something into zero equal parts, and division by zero has no answer.

A fraction is also a division

The fraction bar has a second meaning: divide. The fraction 3/4 is the answer to 3 ÷ 4. Picture 3 identical pizzas shared equally among 4 people. Cut each pizza into quarters, which gives 12 quarters, and hand out 3 quarters to each person. Each person receives 3/4 of a pizza, so 3 ÷ 4 = 3/4 = 0.75. This is why a calculator turns any fraction into a decimal: it divides the numerator by the denominator.

A fraction of a group

Fractions also describe parts of a collection. Suppose 2/5 of the 20 students in a class walk to school. Split the class into 5 equal groups of 4 students. Two of those groups walk, so 8 students walk. The calculation is 20 ÷ 5 = 4, then 4 × 2 = 8. Divide by the denominator to find the size of one part, then multiply by the numerator to count the parts you want.

Worked examples

Example 1. A chocolate bar has 12 squares and you eat 5. You ate 5/12 of the bar, and 7/12 is left, because 12 − 5 = 7 squares remain. The two fractions together make 12/12, the whole bar.

Example 2. What is 3/8 of 40 minutes? One eighth of 40 is 40 ÷ 8 = 5 minutes, so three eighths is 3 × 5 = 15 minutes.

Example 3. Seven friends share 2 litres of water equally. Each gets 2 ÷ 7 = 2/7 of a litre, a little under 0.29 litres.