Key takeaway
To reverse a percentage decrease, recover the original: original = final ÷ (1 − r), where r is the decimal rate of drop. A price of $80 after a 20% fall → 80 ÷ 0.80 = $100. Divide by (1 − r); never just add r back to the final number, or you understate the start.
Scope and assumptions
General arithmetic for a single multiplicative decrease. Sequential or compounded drops need the net method. This site is informational and not official.
Short answer: To reverse a percentage decrease, recover the original: original = final ÷ (1 − r), where r is the decimal rate of drop. A price of $80 after a 20% fall → 80 ÷ 0.80 = $100. Divide by (1 − r); never just add r back to the final number, or you understate the start.
Quick navigation
- Percent decrease → calculate-percent-decrease
- Find the base → how-to-find-base-from-a-percent
- Net change → net-percentage-change
- The calculator → Percentage Calculator
1. The idea
After a decrease of r, the final value is (1 − r) of the original. To get back:
original = final ÷ (1 − r)
2. Worked example
A jacket sells for $80 after a 20% markdown. What was the original?
- r = 0.20, so (1 − r) = 0.80
- original = 80 ÷ 0.80 = $100
Check: 20% of 100 = 20; 100 − 20 = 80. ✓
3. The add-back trap
A common error: "20% off, so add 20% to 80" → 80 × 1.20 = 96.
That is wrong — 20% of 80 is 16, giving 96, not 100. The 20% was taken from the original 100, not from 80. Always divide by (1 − r).
4. Table
| Final | Drop | (1−r) | Original = final ÷ (1−r) |
|---|---|---|---|
| 80 | 20% | 0.80 | 100 |
| 90 | 10% | 0.90 | 100 |
| 75 | 25% | 0.75 | 100 |
| 64 | 20% | 0.80 | 80 |
5. If it was an increase
The mirror formula uses (1 + r):
original = final ÷ (1 + r)
A value of 110 after a 10% rise → 110 ÷ 1.10 = 100.
6. Common mistakes
- Adding r to the final — understates the original (80 + 20% ≠ 100).
- Dividing by r instead of (1 − r) — 80 ÷ 0.20 = 400, wildly wrong.
- Using the wrong sign — a decrease uses (1 − r); an increase uses (1 + r).
- Applying to a compounded drop — two drops need division by the factor product, not a single (1 − r).
7. Reverse an increase too
If a value rose instead of fell, the mirror formula applies: original = final ÷ (1 + r). A price of $110 after a 10% rise came from 110 ÷ 1.10 = $100. The logic is identical to reversing a drop — you divide by the growth factor rather than by (1 − r). In both cases you undo the scaling, you never subtract or add the percent to the final number, which is the error that leaves you short of the true start.
8. The undo is a division
Reversing a decrease is always a division by the remaining factor, because the drop was a multiplication by (1 − r) that you now undo. A $80 price after 20% off returns to 80 ÷ 0.80 = $100. Adding 20% to 80 is the wrong undo because the 20% was taken from 100, not from 80; only division by the factor recovers the true start. The same logic reverses an increase with (1 + r), and in both cases the operation is divide, never add, which is the habit that prevents the understated original from slipping through.
9. Divide, never add
The undo of a decrease is a division by the remaining factor, full stop. A $80 price after 20% off is 80 ÷ 0.80 = $100; adding 20% to 80 gives $96, which is wrong because the 20% was taken from 100, not from 80. The factor (1 − r) is what the drop multiplied by, so only division by it returns the start. Make "divide to reverse" the reflex and the recovered original is always right, with no temptation to patch the number by adding the percent back to the reduced value.
FAQ
Q: How do I reverse a percentage decrease? original = final ÷ (1 − r). A $80 price after 20% off → 80 ÷ 0.80 = $100.
Q: Why not add 20% to $80? Because 20% of 80 is 16, not 20; the drop was 20% of the original 100. You understate the start.
Q: What if it was an increase instead? Then original = final ÷ (1 + r). Same idea, plus the rate.
Q: Can I use the base formula? Yes — this is a special case of find-base-from-a-percent with the rate expressed as a remaining share (1 − r).
Q: What if there were two decreases? Divide by the product of (1−r) factors, or use net-percentage-change logic.
Sources: Investopedia — Percentage Change; Khan Academy percentages.
Disclaimer: General arithmetic for a single multiplicative decrease. Sequential or compounded drops need the net method. This site is informational and not official.
Sources and references
- Investopedia — Percentage Changehttps://www.investopedia.com/terms/p/percentage-change.asp
- Khan Academy percentageshttps://www.khanacademy.org/math/pre-algebra/x31947de81e3f6453:percentages