Skip to content
PiRush

LearnArithmetic

Fractions

Find a fraction of a number, like 3/4 of 60, and add fractions with different bottoms by finding a common denominator.

A fraction is a part of a whole. When you cut a cake into 4 equal pieces and eat 3 of them, you have eaten three quarters of the cake, written 3/4. Fractions are everywhere: half an hour, a third of the price, three quarters of a cup of flour. The word comes from the Latin word for "broken", because a fraction is a whole broken into equal parts. People in ancient Egypt were already writing fractions more than 3,500 years ago.

What the two numbers mean

A fraction has a top number and a bottom number. The bottom number is the denominator. It tells you how many equal parts the whole is cut into. The top number is the numerator. It tells you how many of those parts you take.

A bar cut into 4 equal parts, with 3 of them shaded. The shaded part is 3/4 of the bar.
  • 1/2 is one half: the whole is cut into 2 parts and you take 1.
  • 2/3 is two thirds: the whole is cut into 3 parts and you take 2.
  • 8/8 is all 8 parts out of 8, which is one whole. Any fraction with the same top and bottom is 1.

A fraction of a number

To find 3/4 of 60, first find one part, then take as many parts as the top number says.

  1. Divide the number by the bottom number. This gives one part.
  2. Multiply that part by the top number.

Example 1

What is 3/4 of 60?

  1. One quarter of 60: 60 ÷ 4 = 15.
  2. Three quarters is 3 of those: 3 × 15 = 45.

Answer: 45

Example 2

What is 5/8 of 64?

  1. One eighth of 64: 64 ÷ 8 = 8.
  2. Five eighths: 5 × 8 = 40.

Answer: 40

Equal fractions and simplifying

Different fractions can be the same amount. Half a pizza is the same as 2 slices out of 4, or 3 slices out of 6. So 1/2 = 2/4 = 3/6. These are called equivalent fractions.

Three bars of the same length. The shaded part is the same size in each one.

Remember

Multiply or divide the top and the bottom by the same number, and the fraction keeps its value.

To simplify a fraction means to write it with the smallest numbers you can. Divide the top and the bottom by a number that goes into both. For 6/20, both numbers can be divided by 2, so 6/20 = 3/10. Nothing else goes into both 3 and 10, so 3/10 is as simple as it gets. Always simplify your final answer.

Adding and subtracting fractions

You can only add fractions when the pieces are the same size. 1/5 + 2/5 = 3/5, because one fifth and two fifths make three fifths. The bottom stays the same, and you add the tops.

When the bottoms are different, first change both fractions so they have the same bottom. This shared bottom is called a common denominator.

  1. Find a number that both bottom numbers divide into. Multiplying the two bottoms always works.
  2. Change each fraction to that bottom. Whatever you multiply the bottom by, multiply the top by too.
  3. Add or subtract the tops. Keep the bottom.
  4. Simplify if you can.

Example 3

What is 1/3 + 1/4?

  1. Both 3 and 4 go into 12, so use twelfths.
  2. 1/3 = 4/12 (top and bottom times 4).
  3. 1/4 = 3/12 (top and bottom times 3).
  4. 4/12 + 3/12 = 7/12.

Answer: 7/12

Example 4

What is 2/3 − 1/2?

  1. Both 3 and 2 go into 6, so use sixths.
  2. 2/3 = 4/6 and 1/2 = 3/6.
  3. 4/6 − 3/6 = 1/6.

Answer: 1/6

Example 5

What is 1 − 3/8?

  1. Write the whole 1 as eighths: 1 = 8/8.
  2. 8/8 − 3/8 = 5/8.

Answer: 5/8

Multiplying fractions

Multiplying is easier than adding. You do not need a common denominator.

Remember

a/b × c/d = (a × c)/(b × d)

Top times top, bottom times bottom.

Example 6

What is 3/4 × 2/5?

  1. Tops: 3 × 2 = 6.
  2. Bottoms: 4 × 5 = 20.
  3. So the answer is 6/20.
  4. Divide the top and the bottom by 2: 3/10.

Answer: 3/10

The word "of" means multiply. Half of a third is 1/2 × 1/3 = 1/6. Cut a third of a cake in half and you get a sixth of the cake. This also explains why multiplying two fractions smaller than 1 gives a smaller answer.

Dividing fractions

To divide by a fraction, turn the second fraction upside down and multiply. The upside-down fraction is called the reciprocal: the reciprocal of 5/3 is 3/5.

Remember

a/b ÷ c/d = a/b × d/c

Keep the first fraction. Change divide to times. Flip the second fraction.

Example 7

What is 5/6 ÷ 5/3?

  1. Keep 5/6, change to times, and flip 5/3 to 3/5.
  2. 5/6 × 3/5 = 15/30.
  3. 15 goes into both numbers: 15/30 = 1/2.

Answer: 1/2

Why does flipping work? Think about 3 ÷ 1/2. It asks: how many halves are there in 3? Each whole holds 2 halves, so 3 wholes hold 6 halves. And 3 × 2 = 6. Dividing by 1/2 is the same as multiplying by 2.

All four at a glance

To do thisYou needExample
Add or subtracta common denominator1/2 + 2/5 = 5/10 + 4/10 = 9/10
Multiplytop times top, bottom times bottom2/3 × 1/5 = 2/15
Divideflip the second, then multiply1/4 ÷ 1/2 = 1/4 × 2/1 = 1/2
Take a fraction of a numberdivide by the bottom, times the top2/3 of 45 = 2 × 15 = 30

Try it yourself

  1. 1.What is 2/5 of 35?

    Show the answer

    14

    One fifth of 35 is 35 ÷ 5 = 7, and two fifths is 2 × 7 = 14.

  2. 2.What is 1/2 + 1/3?

    Show the answer

    5/6

    Use sixths: 3/6 + 2/6 = 5/6.

  3. 3.What is 3/4 − 1/3?

    Show the answer

    5/12

    Use twelfths: 9/12 − 4/12 = 5/12.

  4. 4.What is 1 − 2/5?

    Show the answer

    3/5

    One whole is 5/5, and 5/5 − 2/5 = 3/5.

  5. 5.What is 2/3 × 3/4?

    Show the answer

    1/2

    Top times top and bottom times bottom gives 6/12, and 6 goes into both, so it is 1/2.

  6. 6.What is 3/4 ÷ 1/2?

    Show the answer

    3/2, which is 11/2

    Flip the second fraction and multiply: 3/4 × 2/1 = 6/4 = 3/2. There are one and a half halves in three quarters.

Ready to test yourself?

Play Arithmetic: 20 seconds per question, and it gets harder as you go.