Finding the Whole from a Percent

20% of a number is 15. The part is known and the percent is known, but the whole is not.

Go down to 1%

If 20% is 15, then 1% is 15 ÷ 20 = 0.75.

Then up to 100%

100% is 100 × 0.75 = 75. The whole is 75.

Check it

20% of 75 is 15. It fits, so the whole is right.

Working backwards from a part

If 30% of a number is 45, what is the number? Since 30% is 45, 10% is 15, so 100% is 150. Getting to 1% or 10% first and then scaling to 100% is the most dependable route and needs no formula.

The direct route

The number is 45 ÷ 0.30 = 150. Dividing by the decimal form does in one step what the scaling does in two. Notice the answer is larger than 45 — dividing by a number less than one enlarges, which is the Grade 5 rule still holding.

The classic error

People often calculate 30% of 45 instead, getting 13.5. That answers a different question. Before calculating, ask whether the answer should be bigger or smaller than 45. Here the 45 is only part of the total, so the answer must be bigger.

Where it comes up

A shirt is 30% off and costs £42 — what was the original price? £42 is 70% of it, so the original is 42 ÷ 0.7 = £60. Sale prices, tips and tax all require working backwards, and the mistake above is what makes people misjudge a discount.