Problem 15 · 1986 AJHSME
Hard
Fractions, Decimals & Percents
multiply-discount-factors
Sale prices at the Ajax Outlet Store are 50% below original prices. On Saturdays an additional discount of 20% off the sale price is given. What is the Saturday price of a coat whose original price is $180?
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Answer: B — $72.
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Hint 1 of 2
The two discounts don't add up to 70% off. The 20% comes off the *already-reduced* sale price, not the original — so the discounts stack one after the other. What fraction of the price *survives* each cut?
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Hint 2 of 2
Track what you keep, not what you lose: after "50% off" you keep 0.5 of the price; after a further "20% off" you keep 0.8 of that. Multiply the keep-factors.
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Approach: chain the 'fraction kept' factors
- Each discount is taken on the current price, so they multiply rather than add. After 50% off you keep half: $180 × 0.5 = $90 (the sale price).
- On Saturday, 20% off means you keep 80% of *that*: $90 × 0.8 = $72.
- Watch the trap: 50% + 20% is *not* 70% off (that would give $54). Successive discounts stack multiplicatively — keeping 0.5 then 0.8 means keeping 0.5 × 0.8 = 0.4, i.e. 40% of $180.
- Why this transfers: any chain of percent changes is handled by multiplying 'fraction remaining' factors — far safer than adding or subtracting the percents.
Another way — combine the factors first:
- Keep-factor overall = 0.5 × 0.8 = 0.4, so Saturday price = $180 × 0.4 = $72 in one step.
Mark:
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