A jacket marked 20% off costs $72. What was the original price? The tempting move is to add 20% to $72 and call it $86.40. That is wrong, and the receipt makes the error easy to see: 20% of $90 is $18, and $90 − $18 = $72. The original was $90. Reverse percentage undoes a percentage change by dividing, not by adding the rate back. This guide covers the formula for discounts, GST-inclusive totals and raises, with checks you can do in your head.
Part of the Percentage Calculator Guide. Run the division in the percentage calculator, see forward sale math in discount and sale price, and split inclusive tax with the GST calculator.
The reverse percentage formula
Start with the forward equation. A price after a discount is final = original × (1 − r), where r is the discount rate as a decimal. A value after an increase is final = original × (1 + r). To go backward, divide both sides by that factor:
Original after a decrease = final ÷ (1 − r). Original after an increase = final ÷ (1 + r). The rate must be a decimal in the division: 20% becomes 0.20, not 20.
For the jacket: 72 ÷ (1 − 0.20) = 72 ÷ 0.80 = 90. For a salary of $56,000 after a 12% raise: 56,000 ÷ 1.12 = 50,000. The same logic works when you know a part and its share of the whole: whole = part ÷ (rate ÷ 100). If 30 is 20% of a number, the number is 30 ÷ 0.20 = 150.
The common mistake is subtracting the rate from the final value. A 20% discount is 20% of the original, not 20% of the sale price, so the final value cannot serve as the base. The error grows with the size of the rate. At 50% off, a $50 sale price implies a $100 original; subtracting 50% from $50 gives $25, which is not even close. At 10% off, the shortcut lands at $79.20 instead of $80, a small miss that still compounds across a cart.
Check forward, not backward: after finding the original, multiply it by the same factor. If 90 × 0.80 returns 72, the reverse step is correct. This one-line check catches most mistakes.
Worked example: original price before a discount
Take the $72 jacket at 20% off. The forward factor is 1 − 0.20 = 0.80. Divide: 72 ÷ 0.80 = 90. Verify: 90 × 0.20 = 18 discount, and 90 − 18 = 72. The original price was $90 and the shopper saved $18.
A second case: a desk sells for $84 after 25% off. The factor is 0.75. Divide: 84 ÷ 0.75 = 112. Verify: 112 × 0.25 = 28, and 112 − 28 = 84. The original was $112, the saving $28. Notice that $84 is 75% of $112, not 75% of some smaller number; the sale price always represents the remaining share of the original.
Stacked discounts need one extra step. Suppose a $72 final price followed a 20% clearance and then an extra 10% at the register. Each cut applies to the balance left by the previous one, so the combined factor is 0.80 × 0.90 = 0.72. Divide: 72 ÷ 0.72 = 100. Verify forward: 100 → 80 after 20% off → 72 after 10% off. Dividing by 0.70 would give 102.86, which is wrong because the two rates do not add.
| Final amount | What already happened | Reverse factor | Original | Forward check |
|---|---|---|---|---|
| $72 | 20% discount | ÷ 0.80 | $90 | 90 × 0.80 = 72 |
| $84 | 25% discount | ÷ 0.75 | $112 | 112 × 0.75 = 84 |
| ₹1,180 | 18% GST added | ÷ 1.18 | ₹1,000 | 1,000 × 1.18 = 1,180 |
| $56,000 | 12% raise applied | ÷ 1.12 | $50,000 | 50,000 × 1.12 = 56,000 |
One pattern to remember: a discount factor is below 1, so dividing by it makes the number larger; an increase factor is above 1, so dividing by it makes the number smaller. If your answer moves in the wrong direction, you used the wrong sign.
Reverse GST and reverse a raise
A total of ₹1,180 includes 18% GST. The base is not 1,180 − 18% of 1,180, because the tax was calculated on the pre-tax amount. Divide by 1.18: 1,180 ÷ 1.18 = ₹1,000. The tax portion is ₹180, confirmed by 1,000 × 0.18 = 180 and 1,000 + 180 = 1,180. If you subtract 18% from the inclusive total you get ₹967.60 and a tax figure of ₹212.40, both wrong.
Order matters when a discount and a tax meet. A $100 item with 20% off and then 10% sales tax ends at 100 × 0.80 × 1.10 = $88. To reverse a final $88 in one step, divide by 0.88, the product of the two factors. Dividing by 1.10 and then by 0.80 returns the same $100. What you cannot do is subtract 20% and then 10% from the final amount; percentages applied in sequence multiply, so reverse them by dividing by each factor in turn.
The same reversal handles pay raises. A salary of $56,000 after a 12% increase means the old salary was 56,000 ÷ 1.12 = $50,000. The raise was $6,000, which is 12% of $50,000. Subtracting 12% from the new salary gives $49,280 and understates the old salary by $720, because 12% of $56,000 is not the raise that was applied.
This is also where margin and markup get confused. A $100 price with a 20% margin leaves $80 of cost, so you multiply by 0.80. A $100 price built from a 20% markup on cost implies a cost of 100 ÷ 1.20 = $83.33. The words describe different bases, and reversing the wrong one puts your cost off by several dollars per unit.
Five mistakes that produce the wrong original
- Adding the discount back to the sale price. 20% of $72 is $14.40, but the actual discount was 20% of $90, or $18. The final value is the wrong base.
- Subtracting an included tax from the gross total. 18% of ₹1,180 is ₹212.40, while the GST inside the total is ₹180. Always divide by 1 + rate to find the base first.
- Adding stacked rates. 20% then 10% is not 30%. The combined factor is 0.80 × 0.90 = 0.72, so a $72 final price points to a $100 original.
- Rounding before the final step. A $72.99 item after 15% off gives 72.99 ÷ 0.85 = 85.8706, or $85.87. Rounding the sale price to $73 first gives $85.88, a one-cent miss; rounding the factor 0.85 to 0.9 gives $81.10, a four-dollar miss.
- Mixing percent with percentage points. If a rate falls from 5% to 4%, that is 1 percentage point, not a 1% cut. The relative decline is 1 ÷ 5 = 20%. Reversing a points change and reversing a percent change are different operations.
A four-step workflow you can repeat
- Write the forward relationship: final = original × factor. Name whether the change was a discount, tax or raise.
- Turn the rate into a factor: 1 − r for a decrease, 1 + r for an increase. For stacked cuts, multiply the factors first.
- Divide the final amount by the factor. Keep at least four decimals until the last step.
- Multiply the answer forward by the same factor. If it does not match the final amount, recheck the rate or the direction.
For a quick second opinion, enter the original and rate in the percentage calculator and confirm the forward result. For GST-inclusive invoices, the GST calculator extracts the base and tax in one pass, which removes the temptation to subtract the rate from the gross total.