Reverse Percentage: Find the Original Price Before a Discount
You know the price you paid, and you know the discount was 30%. What was it before? This is a reverse percentage problem, and it is the single most commonly botched calculation in everyday maths — because the intuitive move (add 30% back) gives the wrong answer every single time.
Divide, don't add back. Original = Final ÷ (1 − discount as a decimal). A jacket costing $70 after 30% off was 70 ÷ 0.70 = $100, not 70 + 30% = $91. For an increase, divide by (1 + rate): $120 after a 20% markup came from 120 ÷ 1.20 = $100.
Why adding the percentage back fails
Percentages are always taken of something. A 30% discount is 30% of the original price, not 30% of what you paid. When you add 30% to the discounted figure, you are taking the percentage of the wrong base — a smaller number — so you always land short of the true original.
The gap is bigger than people expect. At 30% off, adding back gives $91 instead of $100: a $9 error on a $100 item. At 50% off it is worse — adding 50% to $50 gives $75, when the real original was $100.
Adding the percentage back applies the rate to the smaller number, so it always undershoots the true original.
The two reverse formulas
Everything reduces to one idea: the final amount equals the original multiplied by a factor, so recovering the original means dividing by that factor.
- After a decrease: Original = Final ÷ (1 − rate)
- After an increase: Original = Final ÷ (1 + rate)
A 25% discount means you paid 75% of the original, so the factor is 0.75. A 15% service charge means you paid 115%, so the factor is 1.15. Identify the factor first and the arithmetic is a single division.
Working back from a discount
A laptop is advertised at $884 after 32% off. The factor is 1 − 0.32 = 0.68. So the original is 884 ÷ 0.68 = $1,300, and the saving is $416. Check it forward: 1,300 × 0.32 = 416, and 1,300 − 416 = 884. Always verify by running the calculation forwards — it takes five seconds and catches every sign error.
Stacked discounts work the same way, with the factors multiplied. An item reduced 20% and then a further 10% sits at 0.80 × 0.90 = 0.72 of the original. Paying $216 means the original was 216 ÷ 0.72 = $300. Note this is a 28% total discount, not 30% — see chaining percentage changes.
Removing VAT or sales tax
Tax-inclusive prices are reverse percentages in disguise. With 20% VAT, the shelf price is 120% of the net price, so net = gross ÷ 1.20 and the tax portion is gross − net.
| Gross price | Tax rate | Net price | Tax |
|---|---|---|---|
| $120.00 | 20% | $100.00 | $20.00 |
| $107.50 | 7.5% | $100.00 | $7.50 |
| $54.00 | 8% | $50.00 | $4.00 |
| $1,250.00 | 25% | $1,000.00 | $250.00 |
A common slip is taking 20% of the gross price. On $120 that gives $24 — too high, because the tax was never charged on the tax.
Quick divisor table
| Change applied | Divide final by | Shortcut |
|---|---|---|
| 10% off | 0.90 | × 10 ÷ 9 |
| 20% off | 0.80 | × 1.25 |
| 25% off | 0.75 | × 4 ÷ 3 |
| 50% off | 0.50 | × 2 |
| 10% added | 1.10 | × 10 ÷ 11 |
| 25% added | 1.25 | × 0.8 |
Practice problems
1. You paid $63 after 10% off. What was the list price?
2. A salary is $58,300 after a 6% raise. What was it before?
3. A bill comes to $92 including a 15% service charge. What was the food total?
4. After 20% then 25% off you pay $180. What was the original?
Frequently Asked Questions
Found this useful?
Share it with someone who should start investing today.