How do you add and subtract fractions?

Adding fractions with the same denominator

Meena ate 1/2 of a chikki and her brother ate 1/4 of it. How much did they eat together? Cut the chikki into halves, and cut the remaining half into two quarters. Meena’s half is 2 quarters, and her brother ate 1 quarter, so together they ate 1/4 + 1/4 + 1/4 = 3/4 of the chikki. The 1/4 that is left is what remains.

When the fractions already have the same fractional unit, adding is just counting units:

• 2/5 + 1/5 = 3/5 (2 fifths and 1 fifth make 3 fifths).
• 4/7 + 6/7 = 10/7 = 7/7 + 3/7 = 1 3/7.

On a number line, start at 4/7 and take 6 more hops of 1/7. You land on 10/7, which is 3/7 past 1.

Adding fractions with different denominators

1/4 and 1/3 are parts of different sizes, so we cannot just count them. First turn them into parts of the same size. Both 4 and 3 divide 12:

• 1/4 = (1 × 3)/(4 × 3) = 3/12
• 1/3 = (1 × 4)/(3 × 4) = 4/12

Now 1/4 + 1/3 = 3/12 + 4/12 = 7/12.

Brahmagupta’s method

This method of adding any number of fractions was first described in general by Brahmagupta in 628 CE:

1. Find equivalent fractions with a common fractional unit, using a common multiple of the denominators (their product, or the smallest common multiple).
2. Add the numerators and keep the same denominator.
3. Write the answer in lowest terms if needed.

Example: 2/3 + 1/5. The smallest common multiple of 3 and 5 is 15, so 2/3 = 10/15 and 1/5 = 3/15. The sum is 13/15.

Example: 1/6 + 1/3. The smallest common multiple of 6 and 3 is 6, so 1/3 = 2/6 and the sum is 1/6 + 2/6 = 3/6. In lowest terms, dividing by 3, that is 1/2.

Subtracting fractions

Brahmagupta’s method works for subtraction too. With the same denominator, subtract the numerators: 6/7 − 4/7 = 2/7 (take away 4 of the 6 sevenths).

With different denominators, first make them the same. For 3/4 − 2/3, use 12:

• 3/4 = (3 × 3)/(4 × 3) = 9/12 (multiply by 3, because 4 × 3 = 12)
• 2/3 = (2 × 4)/(3 × 4) = 8/12 (multiply by 4, because 3 × 4 = 12)

So 3/4 − 2/3 = 9/12 − 8/12 = 1/12. Then simplify if you can.

Word problems

• Ravi walks 3/5 km to the bus stop and then 1/4 km from the bus to school. He walks 3/5 + 1/4 = 12/20 + 5/20 = 17/20 km in all.
• A bottle holds 7/8 litre of oil, and 1/2 litre is used. 7/8 − 1/2 = 7/8 − 4/8 = 3/8 litre is left.

Check that you are adding when parts are put together, and subtracting when a part is taken away or when you find a difference.

A pinch of history

The way we write fractions today began in India. The Bakshali manuscript, from about 300 CE, writes a half as 1 above 2. Brahmagupta wrote down the rules for adding and subtracting fractions, and they reached Europe through the Arabs many centuries later. Ancient Egyptians wrote most fractions as sums of different unit fractions, such as 19/24 = 1/2 + 1/6 + 1/8. Can you find three different unit fractions that add up to 1? There is only one way: 1/2 + 1/3 + 1/6 = 1. Before adding, it helps to know how to compare fractions.