Working Out the Original Price From a Sale Price

Why a rise and an equal fall never cancel

The same structure explains a fact people find genuinely surprising: a 10% increase followed by a 10% decrease does not return you to the start.

NRICH's Percentage Unchanged problem makes the point cleanly. If one quantity grows by 10%, so is multiplied by 1.1, the compensating change is multiplication by 1/1.1, which is about 0.909 — a decrease of roughly 9.09%, not 10%. Reversing a multiplier takes its reciprocal, and a reciprocal is not a complement.

The cleanest case: reversing a 25% markup means dividing by 1.25, and 1/1.25 is exactly 0.8 — a 20% reduction. A 25% rise is undone by a 20% fall, and the asymmetry is not an accident of these particular numbers.

The same move on tax and on markup

Every one of these problems is "a number was multiplied by something; undo it".

Sales tax. A tax-inclusive total is the pre-tax price × (1 + rate). At 8.25%, a total of 108.25 came from a pre-tax price of 108.25 ÷ 1.0825 = 100. Subtracting 8.25% from the total would give 99.32, wrong for the same reason as the discount case. The forward version, and why 8.25% decomposes so nicely, is in estimating sales tax from friendly blocks.

Retail markup. If a seller marks up cost by 25%, the price is cost × 1.25, so cost is price ÷ 1.25 — which, as above, is price × 0.8.

Checking a "was / now" claim for internal consistency

Reverse percentages give you a cheap consistency test on advertising. If a sign says "40% off — now 45", the implied original is 45 ÷ 0.6 = 75. If the ticket beside it claims the original was 70, the two statements disagree, because 70 × 0.6 is 42, not 45.

That is an arithmetic observation, not a legal one. In the United States, the Federal Trade Commission publishes a set of guides on deceptive pricing at 16 CFR Part 233, which cover comparisons made against a former price. Whether any particular claim complies is a question for the regulator and the seller; what the arithmetic gives you is the ability to notice when a sign is not internally consistent with itself.

The What Works Clearinghouse practice guide on fractions instruction recommends building conceptual understanding of ratio, rate and proportion strategies before handing learners cross-multiplication as a procedure, and this is a good illustration of why. A reader who understands that the sale price is a proportion of the original will never add the percentage back; a reader who has only memorised a formula has nothing to catch the mistake.

Practising the reverse move

IQ Test & Brain Training Games, which is our own app, built by Reign Creative, ships a Discount Challenge mode — a 75-second timed drill on the Discounts difficulty. Its techniques library states the forward rule plainly: for "X% off", you can subtract the discount, or jump straight to what you pay, so 30% off means you pay 70%. Holding the forward rule in that form is what makes the reverse obvious, because "you pay 70%" is already the multiplier you need to divide by.

The Percentage Trainer covers the same territory over a wider range, with difficulty settings for Basics, Tips, Discounts, Advanced and Mixed. Each technique in the library carries steps, a worked example, a "best for" note, a named common mistake and a two-question mini quiz, and links through to a specific mode at a specific difficulty.

The app is an educational game and a practice tool. Its drills record score, accuracy, streak, response time and personal bests, and those numbers describe drill performance; its IQ-style test and other test results are, in its listing's words, estimates based on app activities, not clinical or professional assessments. The methods that sit around this one are collected in the mental math training guide, the rest of the hub under mental math, and the wider subject under education and brain training. For related percentage shortcuts including near-100 multiplication and successive percentage change, see mental math tricks that actually work.