When the final value is known and you need to work backward to find where it came from, the key idea is simple: you do not undo a discount by adding the same percentage, and you do not undo an increase by subtracting the same percentage from the final value. The original percentage was calculated from the original amount, so the base changes.
The safest method is to turn the percentage change into a multiplier. With a 20% discount, the final value is 80% of the original, or 0.8 times the original. With a 20% increase, the final value is 120% of the original, or 1.2 times the original. If you know the final value, divide by that multiplier.
The reverse-percentage rule
For a rate written as a decimal, use:
- discount of r: final value = original value × (1 − r);
- increase of r: final value = original value × (1 + r).
Rearranging the formulas:
- after a discount: original value = final value ÷ (1 − r);
- after an increase: original value = final value ÷ (1 + r).
| Situation | Multiplier applied to the original | How to recover the original |
|---|---|---|
| 10% discount | 0.90 | final ÷ 0.90 |
| 25% discount | 0.75 | final ÷ 0.75 |
| 10% increase | 1.10 | final ÷ 1.10 |
| 35% increase | 1.35 | final ÷ 1.35 |
This reasoning follows directly from standard discount and markup formulas. OpenStax, for example, gives the sale price as original price × (1 − discount rate) and the retail price after a markup as cost × (1 + markup rate). “Reverse percentage” is simply the inverse algebraic operation applied to those relationships. See Discounts, Markups, and Sales Tax.
Why adding the percentage back gives the wrong answer
Suppose an item originally costs $100 and receives a 20% discount. It now costs $80. If someone tries to recover the original price by increasing $80 by 20%, the calculation is:
$80 × 1.20 = $96.
It does not return to $100 because the two 20% calculations use different bases. The original discount was 20% of $100, which is $20. The attempted “increase back” is 20% of $80, which is $16. To return to the original, recognize that $80 represents 80% of the previous price:
$80 ÷ 0.80 = $100.
Example 1: final price of $80 after a 20% discount
Step 1 — turn the discount into a multiplier. A 20% discount leaves 80% of the value: 1 − 0.20 = 0.80.
Step 2 — divide the final value by the multiplier. $80 ÷ 0.80 = $100.
Step 3 — check by going forward. $100 × 0.80 = $80.
So the original price was $100.
Example 2: final value of $250 after a 25% increase
Step 1 — build the increase multiplier. 1 + 0.25 = 1.25.
Step 2 — divide the final value. $250 ÷ 1.25 = $200.
Step 3 — verify. $200 × 1.25 = $250.
Therefore, the original value was $200.
Example 3: a quantity ends at 176 after a 12% reduction
The same logic applies to more than prices. If inventory, a measurement, or another quantity falls by 12% and ends at 176 units, the remaining multiplier is:
1 − 0.12 = 0.88.
Then:
176 ÷ 0.88 = 200.
Check: 200 × 0.88 = 176. The starting quantity was 200 units.
A decision process you can use without memorizing tricks
- Identify the direction: was there a discount/decrease or an increase/markup?
- Convert the rate to a decimal: 18% = 0.18; 7.5% = 0.075.
- Build the multiplier: use 1 − rate for a decrease and 1 + rate for an increase.
- Divide the final value by the multiplier.
- Check: multiply the original value you found by the same multiplier and confirm that it returns to the stated final value.
How to tell quickly whether the answer makes sense
A simple size check catches many mistakes:
- after a discount, the original value must be greater than the final value;
- after an increase, the original value must be less than the final value.
Another check is to redo the forward transformation. OpenStax recommends checking whether a percentage answer makes sense in context; that habit is especially useful in reverse problems. See also Solve Percent Applications.
Common mistakes
- Adding the discount to the final value: the original rate was applied to a different base.
- Subtracting the increase from the final value: for the same reason, the base has changed.
- Dividing by the percentage itself: with a 20% discount, the correct divisor is 0.80, not 0.20.
- Using 80 instead of 0.80: percentages must be converted to decimal form when they are used in the multiplier.
- Rounding too early: keep enough decimal places during the calculation and round only the final result when needed.
What if there are two or more changes in a row?
When changes happen sequentially, the multipliers are also applied sequentially. A 10% increase followed by a 10% discount, for example, gives a multiplier of 1.10 × 0.90 = 0.99. The final value is therefore 99% of the original, not 100%.
If the final value is known and you want to recover the starting value, divide by the product of the multipliers. This is the same reverse-percentage logic applied to a chain of changes.
Practical summary
To find the original value, determine what fraction of the original remained or was created by the percentage change. A 20% discount means 80% remains, so the multiplier is 0.80. A 20% increase means the final value is 120% of the original, so the multiplier is 1.20. In both cases, you return to the original by dividing.
In formula form: original = final ÷ multiplier. The best check is always to go forward again: multiply your answer by the multiplier and confirm that it reproduces the stated final value.
