Problem 9 · 2022 AMC 8
Easy
Fractions, Decimals & Percents
percent-multiplier
A cup of boiling water (212°F) is placed to cool in a room whose temperature remains constant at 68°F. Suppose the difference between the water temperature and the room temperature is halved every 5 minutes. What is the water temperature, in degrees Fahrenheit, after 15 minutes?
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Answer: B — 86°F.
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Hint 1 of 2
The temperature itself isn't what halves — the gap between the water and the room is. Track that gap, not the thermometer.
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Hint 2 of 2
Starting gap: 212 − 68 = 144. In 15 minutes it halves three times. Add the shrunken gap back onto room temperature.
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Approach: track the gap (it's what halves), then add room temp back
- Insight: “halved” describes the difference from room temperature, not the reading itself. So work with the gap. Starting gap: 212 − 68 = 144°F.
- 15 minutes is three 5-minute steps, so halve the gap three times: 144 → 72 → 36 → 18.
- The water is now 18° above the room: 68 + 18 = 86°F.
- You'll see this again: whenever something decays “toward” a fixed level, shift your view to the gap from that level — the gap shrinks by a clean ratio while the raw quantity doesn't.
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