After a 20% discount, the price needs to rise by 25% to return to its original value. The discount is measured against the original price, while the increase is measured against the lower sale price.
Reverse percentages are part of the 2027 SEC G3 Mathematics K310 syllabus. This guide shows how to find the recovery percentage from a pair of prices, from a discount rate alone, or from a price that has only partly recovered.
- Identify the price before the increase.
- Find the amount needed to return to the original price.
- Divide that amount by the price before the increase.
- Check by applying both multipliers in order.
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Find the amount and choose its base
An item originally costs $150. A 20% discount removes $30, leaving $120. The shop then adds back $30. The cash amounts match because the second change restores the first, but the percentages use different starting prices.
Divide the increase of $30 by the sale price of $120. The fraction is 0.25, so the increase is 25%. Check the result by multiplying $120 by 1.25 to get $150.
If you use $150 as the denominator for the increase, you recover the original discount rate. Before dividing, write one line naming the base of the calculation. Here it is the price immediately before the rise.
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Solve when only the discount rate is given
Suppose the question gives a 35% discount and asks for the percentage increase needed to restore the price. Choose a starting value of $100 to make the percentages easy to read. The sale price is $65, and the amount to restore is $35.
The required increase is (35 ÷ 65) × 100% = 53.8% to three significant figures. You can use the same reasoning with any original price because both the amount removed and the sale price scale together.
For a general discount of d%, with 0 < d < 100, use d ÷ (100 − d) and express that fraction as a percentage. A 50% discount needs a 100% increase, since doubling the remaining half restores the whole. A full 100% discount leaves zero as the base, so no finite percentage increase can restore the old price.
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Handle a partial recovery
An item falls from $200 to $160 and then rises to $176. What percentage increase is still needed? The remaining gap is $24 and the current base is $176. The required increase is (24 ÷ 176) × 100% = 13.6% to three significant figures.
Check using multipliers. From $200 to $160 the multiplier is 0.80, and from $160 to $176 it is 1.10. The combined multiplier is 0.88. To return from $176 to $200, use the reciprocal 1 ÷ 0.88, which is about 1.13636.
Try another one. A $240 item loses 25% of its price. A student suggests raising the sale price by 25%. That produces $180 × 1.25 = $225. A $60 rise on $180 is needed, so the actual recovery rate is 33⅓%.
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CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Does a rise followed by an equal percentage fall cancel? For a positive rate less than 100%, the combined multiplier is below 1. A 20% rise then a 20% fall gives 1.20 × 0.80 = 0.96.
How is this different from finding an unknown original price? If you know the sale price and the discount, divide the sale price by the discount multiplier to find the original price. Here the question asks for the rate of the subsequent rise.
What should I show in my working? Show the sale price, the gap to the original price and the ratio using the sale price as its denominator.
Official syllabus and next reading
This independent study guide uses original practice examples. See the official 2027 SEC G3 Mathematics K310 syllabus.