How do you find a percentage of an amount and how do you find the whole?

Three questions, not one

Almost every percentage problem is one of three. You know the amount and want a part of it; you know a part and its percentage and want the whole amount; or you know two amounts and want the ratio between them as a percentage.

Identifying which matters more than the arithmetic. Far more wrong answers come from solving the wrong question than from a slip in calculation, so it is worth saying out loud which of the three it is before writing anything.

A quick way to tell: if the large amount is given, you are looking for a part of it, so multiply. If only a small number is given together with the percentage it represents, the large amount is the unknown, so divide. And if two numbers are given with no percentage at all, you are looking for the ratio.

A percentage of an amount

Convert the percentage to a decimal and multiply. Twenty per cent of eighty is 0.2 times 80, which is sixteen.

You can also go via ten per cent, which is convenient mentally: ten per cent of eighty is eight, so twenty per cent is sixteen. Same answer, no calculator.

Finding the whole amount

Here the direction reverses, so you divide rather than multiply. If twelve is twenty per cent of something, divide twelve by 0.2 and get sixty.

An easy check tells you whether the answer is sensible: the whole amount must come out larger than the part, as long as the percentage is under a hundred. A smaller answer means you multiplied where you should have divided.

Worked examples

1. What is 15 per cent of 200?

   1. Convert: 15 per cent is 0.15
   2. Multiply: 0.15 times 200
   3. Or mentally: 10 per cent is 20, and 5 per cent is 10

   Answer: 30

2. A class has 7 pupils absent, which is 25 per cent of the class. How many pupils are there?

   1. The whole amount is the unknown, so divide
   2. 25 per cent is 0.25
   3. Divide: 7 over 0.25
   4. Check: the answer is larger than 7, as required

   Answer: 28 pupils

3. What percentage is 18 out of 60?

   1. Divide the part by the whole: 18 over 60
   2. This gives 0.3
   3. Multiply by a hundred

   Answer: 30 per cent

Common mistakes

Multiplying when you should divide

Multiplication finds a part of a known whole. When the whole is the unknown the operation reverses, and multiplying produces an answer smaller than the part itself — a clear sign the direction flipped.

Dividing by the wrong number in a ratio question

Divide the part by the whole, not the other way round. The reverse gives a figure over a hundred per cent when the part is smaller than the whole, which is impossible.

Forgetting to multiply by a hundred in a ratio question

The division gives a decimal, not a percentage. Stopping there returns 0.3 instead of thirty per cent — the right answer in the wrong form.

More in Decimals and Percentages

• What do the digits after the decimal point mean?
• How do you compare two decimal numbers?
• How do you add and subtract decimal numbers?
• What happens when you multiply a decimal by ten or a hundred?
• How do you multiply two decimals and where does the point go?
• What is a percentage and how do you convert between percentages, fractions and decimals?
• How do you work out a price after a discount or a price rise?