Reverse Percentages
A teen lesson on reverse percentages: find the original amount before a percentage increase or decrease by dividing by the multiplier, with worked examples, a pitfall warning and a quiz.
Key takeaways
- A reverse percentage finds the original amount before a percentage change was applied.
- Identify the multiplier, then divide the final amount by it to undo the change.
- The final amount represents more than 100% after an increase and less than 100% after a decrease.
- You cannot simply take the same percentage off the final figure — that gives the wrong answer.
Working backwards
In a normal percentage problem you know the original and apply a change. A reverse percentage flips this: you are given the final amount after a change and must find the original.
Real examples:
- A jacket costs $63 in a 30% sale — what was the full price?
- A bill is $84 including 20% tax — what was the price before tax?
- A salary is now $2,750 after a 10% rise — what was it before?
The key skill is undoing a multiplier.
The multiplier idea, reversed
From percentage increase and decrease, a change is applied by multiplying by a multiplier:
- 20% increase → multiply original by 1.20
- 30% decrease → multiply original by 0.70
To go forwards you multiply. To go backwards, you do the opposite: divide by the multiplier.
original × multiplier = final, so original = final ÷ multiplier.
Worked example 1: reversing an increase
After a 20% increase a price is $120. Find the original.
- Multiplier for a 20% increase = 1.20.
- The $120 represents 120% of the original.
- original = 120 ÷ 1.20 = $100.
Check forwards: 100 × 1.20 = 120 ✅.
Worked example 2: reversing a decrease
A coat costs $63 after a 30% discount. Find the original price.
- Multiplier for a 30% decrease = 0.70.
- The $63 represents 70% of the original.
- original = 63 ÷ 0.70 = $90.
Check forwards: 90 × 0.70 = 63 ✅.
Worked example 3: removing tax
A bill is $84 including 20% tax. Find the price before tax.
- The pre-tax price was increased by 20%, so multiplier = 1.20.
- The $84 is 120% of the pre-tax price.
- before tax = 84 ÷ 1.20 = $70.
The tax itself is 84 − 70 = $14.
The classic trap
It is tempting to "just take the percentage back off the final figure". This is wrong.
For Worked example 1, taking 20% off $120:
- 20% of 120 = 24, giving 120 − 24 = $96 ❌ (the correct answer is $100).
Why the error? The original 20% increase was 20% of the smaller original ($100) = $20. But 20% of the larger final ($120) is $24 — a different base, so it over-corrects. Always divide by the multiplier instead.
A summary table
| Situation | What the final represents | Multiplier | Find original by |
|---|---|---|---|
| after a 20% increase | 120% | 1.20 | final ÷ 1.20 |
| after a 25% increase | 125% | 1.25 | final ÷ 1.25 |
| after a 15% decrease | 85% | 0.85 | final ÷ 0.85 |
| after a 30% decrease | 70% | 0.70 | final ÷ 0.70 |
| including 20% tax | 120% | 1.20 | final ÷ 1.20 |
A second method: unitary
If decimals feel uncertain, use the unitary method with percentages:
A coat costs $63 after a 30% discount.
- $63 = 70% of the original.
- So 1% = 63 ÷ 70 = $0.90.
- And 100% = 0.90 × 100 = $90.
Same answer, broken into steps you can see.
Try it yourself
Solve these by dividing by the multiplier, then check forwards:
- After a 10% increase a phone bill is $66. Find the original.
- A TV costs $340 after a 15% discount. Find the original price.
- A meal is $46 including 15% service charge. Find the cost before the charge.
(Answers: $60; $400; $40.)
Why this matters
Reverse percentages appear in tax, VAT, original prices in sales, depreciation and exam questions that catch people out. Mastering the "divide by the multiplier" rule keeps you from the classic trap and links directly to percentages in real life.
Quick quiz
Test yourself and earn XP
After a 20% increase a price is $120. Which calculation finds the original?
The original was multiplied by 1.20, so divide to undo it: 120 ÷ 1.20 = 100.
A coat costs $63 after a 30% discount. What was the original price?
$63 is 70% of the original, so divide by 0.70: 63 ÷ 0.70 = $90.
A bill is $84 including 20% tax. What was the price before tax?
$84 is 120% of the pre-tax price, so 84 ÷ 1.20 = $70.
Why is taking 20% off $120 the wrong way to reverse a 20% increase?
The 20% increase was 20% of the smaller original, but taking 20% off $120 is 20% of the larger final figure, so it over-corrects.
After a 25% increase a value is 250. What was it originally?
250 ÷ 1.25 = 200.
FAQ
A reverse percentage problem gives you the amount after a percentage increase or decrease and asks for the original amount before the change. You work backwards by dividing by the multiplier.
Write the multiplier for the change (1.20 for a 20% increase, 0.70 for a 30% decrease), then divide the final amount by that multiplier. For example, $90 after a 10% increase came from 90 ÷ 1.10 = $81.82.
Because the percentage change was taken from the original, smaller amount, not the final one. Taking the percentage off the final figure uses the wrong base and gives an answer that is close but incorrect.
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