Reverse Percentages

Grade 5

What even is a reverse percentage?

Normal percentage questions give you the starting amount and ask for the new one. Reverse percentage questions flip that: you're given the amount after a change, and you have to find where it started.

The classic trap is to just take the percentage off the number you're given. Don't. That number isn't the original — so a percentage of it isn't the right size. We need a proper method.

The one idea that makes it click

Every reverse percentage question comes down to this: the amount you're given represents a certain percentage of the original. Your job is to get back to 100%.

Find what 1% is worth first. Once you know 1%, you can build up to 100% — which is the original amount.

To find the percentage your number represents:

A rise of 20% means your number is 100 + 20 = 120% of the original. A fall of 20% means it's 100 − 20 = 80%.

Worked example — an increase

EXAMPLE

A jacket costs £60 in the sale after a 20% price rise from the shop. What was the original price?

The £60 is after a 20% rise, so it's 120% of the original. Find 1%: 60 ÷ 120 = £0.50. Now scale up to 100%: 0.50 × 100 =£50. Quick check: 20% of £50 is £10, and £50 + £10 = £60. ✓

Worked example — a decrease

EXAMPLE

After a 15% discount, a game costs £34. What was the price before the discount?

A 15% discount means £34 is 85% of the original. Find 1%: 34 ÷ 85 = £0.40. Scale up: 0.40 × 100 =£40. Check: 15% of £40 is £6, and £40 − £6 = £34. ✓

Don't get caught out

If a question says a price "increased by 20% to £60", the £60 is the new price — that's a reverse percentage. But if it says "increased by 20%, what is the new price" and gives you the original, that's a normal forward one. Read which amount they've actually handed you.

Try this one

A house was valued at £\( V \) in 2018.

Its value increased by 15% in 2019, and then increased by a further 4% in 2020.

In 2021, the value of the house was £218 400.

(a) Show that the value of the house at the start of 2019 was £180 000. [3]

(b) By what percentage did the value increase overall from the start of 2019 to the end of 2021? [2]

Nova's hint:

Part (a) requires reverse compound percentage — undo two successive increases to find the original value.

Ask yourself: 'What combined multiplier represents a 15% increase then a 4% increase, and how do I reverse it?' For part (b), use the overall multiplier directly to find the percentage increase from the original.

Try to Solve it with Nova?