Scope and assumptions
Final = Original × (1 + rate); to reverse, divide the final by (1 + rate). Single increase. Illustrative.
A price climbs 20% to $120 and you want to know what it was before. The tempting move is to knock 20% off $120 — but that's wrong, because the 20% was taken of the original, not of $120. To undo it you divide by 1.20, never subtract.
Recovering the before
Original = Final ÷ (1 + rate)
Because Final = Original × (1 + rate), reversing means dividing, not subtracting.
| Final | Increase | Original |
|---|---|---|
| $120 | 20% | 120 ÷ 1.20 = $100 |
| $150 | 50% | 150 ÷ 1.50 = $100 |
| $88 | 10% | 88 ÷ 1.10 = $80 |
| $200 | 25% | 200 ÷ 1.25 = $160 |
Why divide, never subtract
If you subtract 20% of $120 (=$24), you get $96 — wrong, because the original was $100. The 20% hike was on $100, giving $120; reversing must recover $100, and only 120 ÷ 1.20 does that. Subtracting the percent of the final value uses the wrong base.
The mirror: reversing a decrease
- Increase: Original = Final ÷ (1 + r)
- Decrease: Original = Final ÷ (1 − r)
A $80 item after 20% off → 80 ÷ 0.80 = $100 original. Same logic, different sign inside the parentheses.
Undoing a chain at once
A value rose 10% then 20% to reach $132. Reverse in one step: 132 ÷ [1.10 × 1.20] = 132 ÷ 1.32 = $100. Dividing by the full product undoes the whole chain at once.
Reverse-Engineering a Rise Without the Base
- Subtracting the percent of the final — use divide by 1+r.
- Using 0.20 instead of 1.20 — the divisor must include the 1.
- Applying the decrease formula to an increase — 1−r vs 1+r.
- Forgetting r is decimal — 20% = 0.20.
- Rounding the original before checking — verify by re-adding: 100×1.20 = 120.
- Ignoring the order in multi-step reversals — divide by the product of all factors.
Related advanced operations
For the rest of the advanced percentage operations, start at the hub: Percentage Difference Formula. Adjacent: Continuous Percentage Growth and Percentage Error Calculation.
FAQ
What if I only know the final and the increase amount in dollars, not the percent? Then Original = Final − amount directly (the dollar gap is already the absolute change). You only need the division formula when the change is given as a percent of the original.
How do I reverse a decrease to find the pre-sale price? Mirror formula: Original = Final ÷ (1 − r). An $80 item after 20% off → 80 ÷ 0.80 = $100. Same logic, the factor inside is (1 − r) instead of (1 + r).
Can I reverse if the percent was of the final, not the original? Rare, but if a label says "20% of the final price is tax," then the pre-tax amount is Final × (1 − 0.20) = Final × 0.80 — subtraction works because the percent was taken on the final. Always identify the base.
What if there were fees or rounding on the final? Real reversals can drift because of rounding or fixed fees (e.g., a $2.99 fee isn't a percent). Divide by the pure percent factor, then account for any flat additions separately: Final = Original × (1+r) + fee.
How do I check my reversed answer? Re-apply the increase: Original × (1 + r) should equal the final. If 100 × 1.20 = 120, the reversal is correct. It's a built-in verification step.
Sources: Khan Academy, Percent word problems; Maths is Fun, Reverse percentages.
Disclaimer: Illustrative. For stacked or conditional increases, verify each step's base. Informational only.
Sources and references
- Khan Academy — Percent word problemshttps://www.khanacademy.org/math
- Maths is Fun — Reverse percentageshttps://www.mathsisfun.com/numbers/percentage-reverse.html
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