How to find the original value after a percentage change

In a reverse percentage question, the original amount represents 100%. The amount you know is the result after the change. Write these labels before doing arithmetic: original unknown; final 72; decrease 20%.

After a 20% decrease, 80% of the original remains. As a decimal, 80% is 0.8, so the relationship is original × 0.8 = 72. Dividing both sides by 0.8 gives original = 72 ÷ 0.8 = 90. Treat these prices as a hypothetical example, not a current offer.

For an increase of r%, the multiplier is 1 + r/100. For a decrease of r%, it is 1 − r/100. The forward calculation is final = original × multiplier, so the reverse calculation is original = final ÷ multiplier, provided the multiplier is not zero.

Suppose a club has 138 members after a 15% increase. The final membership is 115% of the original, so the multiplier is 1.15. The original membership was 138 ÷ 1.15 = 120. Check by calculating 15% of 120, which is 18, then adding it to 120.

Do not divide 138 by 0.15. That would treat the final total as the increase alone. If a question instead says that the increase itself was 18 members, then original × 0.15 = 18 is the appropriate relationship.

Adding 20% to the discounted price of 72 gives 86.4, not 90. The added amount is 20% of 72, which is 14.4, while the original discount was 20% of 90, which is 18. Equal percentage numbers do not guarantee equal amounts when their bases differ.

A quick size check helps: before a discount, an ordinary positive price must have been larger than the final price. Before an increase, a positive original total must have been smaller. If your result goes the other way, revisit the multiplier and division.

Suppose an amount is increased by 10% and then decreased by 20%, ending at 88. The forward relationship is original × 1.10 × 0.80 = 88. Work backwards by dividing 88 by 0.80 to get 110, then dividing 110 by 1.10 to get 100.

You can also divide by the combined multiplier: 1.10 × 0.80 = 0.88, and 88 ÷ 0.88 = 100. Do not subtract the percentage figures and call this a 10% decrease. The second change applies to the amount after the first change.

Keep the multipliers unrounded during the calculation. If the question asks for a price to two decimal places, round the final original price at the end rather than rounding each intermediate result.

A 100% decrease leaves zero whatever positive amount you started with. Its multiplier is zero, so dividing the final value by it cannot recover an original. You need more information.

A rounded final value can also limit precision. For example, if 72 is a price rounded to the nearest whole unit rather than an exact price, 90 is a possible original under a 20% discount, but the original is not uniquely determined to the cent. Follow the precision stated in the question.

Do not confuse a percentage change with a percentage-point change. Moving from 40% to 50% is an increase of 10 percentage points and a relative increase of 25%. Read the wording before assigning a multiplier.

Copy this template: original represents 100%; final amount = ___; increase or decrease = ___%; final percentage of original = ___%; multiplier = ___; original = final ÷ multiplier = ___; forward check = ___.

Try this: a bag costs 63 after a 30% discount. The final percentage is 70%, so the multiplier is 0.70. Original = 63 ÷ 0.70 = 90. Check: 30% of 90 is 27, and 90 − 27 = 63.

For an increase example, a quantity is 212 after rising by 6%. Original = 212 ÷ 1.06 = 200. Multiplying 200 by 1.06 returns 212. Writing that forward check catches a reversed operation before you finish.