Problem 22 · 2019 AMC 8
Hard
Fractions, Decimals & Percents
percent-multiplierdifference-of-squares
A store increased the original price of a shirt by a certain percent and then decreased the new price by the same amount. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?
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Answer: E — 40%.
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Hint 1 of 2
Up then down by the same percent does NOT return to the start — the decrease acts on a bigger price. Write it as multipliers: ×(1+p) then ×(1−p), and let difference-of-squares simplify the product.
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Hint 2 of 2
(1+p)(1−p) = 1 − p2. Set that equal to 0.84 and the percent pops right out.
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Approach: the two changes multiply to 1 − p²
- Raising by p then lowering by p multiplies the price by (1 + p)(1 − p) = 1 − p2 — a difference of squares, neatly collapsing the two steps into one.
- Set 1 − p2 = 0.84, so p2 = 0.16 and p = 0.4 = 40%.
- Why this transfers: a percent up and the same percent down always leaves 1 − p2 — strictly less than the original, since the drop applies to a larger amount. Recognizing (1+p)(1−p) as a difference of squares is the shortcut.
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