Problem 22 · 1988 AJHSME
Stretch
Fractions, Decimals & Percents
compound-percent
Tom's Hat Shoppe increased all original prices by 25%. Now the shoppe is having a sale where all prices are 20% off these increased prices. Which statement best describes the sale price of an item?
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Answer: E — The sale price is the same as the original price.
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Hint 1 of 2
The 20% off isn't taken off the original price — it's taken off the *already-raised* price, so the two percents can't simply cancel. Turn each change into a 'multiply by' factor instead.
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Hint 2 of 2
Up 25% means ×1.25; then 20% off means ×0.80. Multiply those two factors together — what do you get?
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Approach: turn each percent change into a multiplier and multiply
- Raising by 25% multiplies the price by 1.25; taking 20% off multiplies by 0.80. Doing both means 1.25 × 0.80 = 1.00 — so the final price equals the original. The sale price is the same as the original price.
- Concrete check: start at $100 → up 25% → $125 → 20% off $125 is $25 off → $100. Right back where we started.
- Trap to avoid: +25% then −20% does NOT make +5%. The 20% is a slice of the larger $125, not of the original $100, so the bigger 'off' exactly undoes the increase. Percent changes combine by multiplying their factors, never by adding the percents.
Mark:
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