Problem 12 · 2004 AMC 8
Medium
Fractions, Decimals & Percents
battery-fraction-rates
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last?
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Answer: B — 8 more hours.
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Hint 1 of 2
Think of the battery as one whole tank, and each mode as a different drain speed. 'Lasts 24 hours idle' means idling drains 1/24 of the tank per hour; 'lasts 3 hours in use' means using drains 1/3 per hour. Split the 9 hours into how much was each mode.
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Hint 2 of 2
The technique is fraction-of-the-job per hour (the same as combined-work-rate problems): convert each 'lasts T hours' into a rate of 1/T per hour, add up what's been spent, and see what's left.
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Approach: track the tank as fractions
- The 9 hours split as 1 hour of talking + 8 hours idle (60 min used). Battery spent = 1 × 1/3 (talking) + 8 × 1/24 (idle) = 1/3 + 1/3 = 2/3 of the tank.
- So 1/3 of the battery remains, and from here on she's idle only, draining 1/24 per hour. Time left = (1/3) ÷ (1/24) = (1/3) × 24 = 8 more hours.
- Intuition check: that one hour of talking ate 1/3 of the battery — as much as 8 whole hours of idling. That's why a single hour of use is so costly here, and a clean way to feel the 1/3-vs-1/24 gap.
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