Reverse Percentages

You spot a jacket in a "20% off" sale. The tag on the rail says \pounds 60. Out of curiosity — what was it selling for last week, before the sale?

Almost everyone's first instinct is to add 20% back on: 60 + 20\% = \pounds 72. That is wrong, and it's a genuine money mistake — the kind that trips up adults at the till, catches out shopkeepers working out their own prices, and quietly loses businesses real cash. The true answer is \pounds 75, and the gap between the two is £3 of pure confusion.

The reason is subtle and worth getting straight once and for all: the 20% that was knocked off was 20% of the larger original price, not 20% of the reduced £60. To travel backwards from a final amount to the original, you need reverse percentages.

Work forwards first, then undo it

A percentage change is really a multiplication. Taking 20% off means keeping 80% — so the shop did this:

\text{original} \times 0.80 = \pounds 60

The £60 is the output of a multiply by 0.80. To get back to the input, you undo a multiply with a divide:

\text{original} = 60 \div 0.80 = \pounds 75

And that checks out: 20% of £75 is £15, and 75 - 15 = 60. The maths comes home. Notice that adding 20% back gave £72 because 20% of the smaller £60 is only £12 — the wrong-sized slice, taken off the wrong base.

The whole method is one idea: treat the amount you know as a percentage of the original, turn that percentage into a multiplier, and divide.

The rule

To find the original amount before a percentage change:

identify the multiplier — a 20% rise gives 1.20, a 25% fall gives 0.75;
the amount you know is \text{original} \times \text{multiplier};
so reverse it by dividing: \text{original} = \text{known} \div \text{multiplier}.

The common mistake is to add or subtract that percentage of the new amount — that uses the wrong base. Always divide.

See it on a bar

Here is a smaller example on a bar. We pay \pounds 40 after a 20% discount, so that £40 fills only 80% of the whole. Find what one 20% slice is worth, then build back up to the full 100%.

In one step: 40 \div 0.80 = 50. The original price was \pounds 50. The bar shows why — 80% is £40, so each 20% slice is £10, and five slices make £50.

Worked example 1 — before a discount

A pair of trainers costs \pounds 68 in a 15% off sale. What was the original price?

15% off means 85% remains, so the multiplier is 0.85.
Divide: 68 \div 0.85 = \pounds 80.
Check: 15% of £80 is £12, and 80 - 12 = 68. Correct.

Worked example 2 — stripping the VAT out of a bill

A plumber's bill comes to \pounds 240, and that figure already includes 20% VAT. How much was the work itself, before tax?

Adding 20% VAT multiplies the pre-tax price by 1.20.
So strip it out by dividing: 240 \div 1.20 = \pounds 200.
Check: 20% of £200 is £40, and 200 + 40 = 240. Correct.

Note the trap: taking 20% off £240 gives £48, so £192 — which is wrong, because the VAT was 20% of the smaller £200, not of the £240 total.

Worked example 3 — before a pay rise

After a 5% pay rise, someone earns \pounds 31{,}500 a year. What was their salary before the rise?

A 5% rise multiplies the old salary by 1.05.
Reverse it: 31{,}500 \div 1.05 = \pounds 30{,}000.
Check: 5% of £30,000 is £1,500, and 30{,}000 + 1{,}500 = 31{,}500.

Spotting the multiplier fast

The whole method lives or dies on one step: turning the percentage into the multiplier the original was multiplied by. Read the sentence carefully and ask "what fraction of the original is left?"

"15% off" → 85% remains → divide by 0.85.
"includes 20% VAT" → 120% of the pre-tax price → divide by 1.20.
"after a 40% discount" → 60% remains → divide by 0.60.
"after a 5% pay rise" → 105% of the old salary → divide by 1.05.

Increases give a multiplier bigger than 1; decreases give one smaller than 1. Once you have it, the answer is a single division — every reverse-percentage question, no matter the story, collapses to \text{known} \div \text{multiplier}.

You cannot reverse a percentage change by applying the same percentage the other way. A £60 jacket that has had 20% taken off is not 60 + 20\% = \pounds 72 — it is 60 \div 0.80 = \pounds 75.

The reason: the 20% discount was 20% of the original (larger) price, but adding "20% back" takes 20% of the reduced price — a smaller number, so a smaller slice. Up-then-down by the same percentage never lands you back where you started. Always identify the multiplier and divide; never just tack the percentage back on.

Reverse percentages are exactly how you strip sales tax (VAT) out of a total. Shops show you tax-inclusive prices — the number on the shelf already has the VAT baked in — but the business has to report the pre-tax figure to the tax office and to itself. Every till receipt that splits out "VAT: £x" is running a reverse percentage in the background.

The same move reveals a shop's original markup: if a coat sells for £75 after a "20% off" sign, you now know the "real" price the shop set was… also worth a raised eyebrow. It's a genuinely useful life skill — one that lets you see through a sale sticker and check whether the deal is as good as it looks.