Problem 25 · 2023 Math Kangaroo
Stretch
Ratios, Rates & Proportions
unit-rate
When Matilda’s smartphone is fully charged it has a battery life of 32 hours if she phones continuously, 20 hours if she surfs the internet continuously, and 80 hours if she does not use it at all. Matilda boards a train with a half-full battery. During her time on board she spends the same amount of time each on phoning, surfing the internet, and not using the phone at all. Just when she arrives at her destination the battery is empty. How many hours did the train ride take?
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Answer: D — 16
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Hint 1 of 2
Convert each usage into a fraction of the battery drained per hour.
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Hint 2 of 2
With equal time on each activity, set the total drain equal to half a battery.
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Approach: add the drain rates over equal time shares
- Drain rates are 1/32, 1/20, 1/80 of the battery per hour.
- Over a ride of length T with T/3 on each: (T/3)(1/32+1/20+1/80) = 1/2.
- The bracket is 15/160 = 3/32, so (T/3)(3/32) = T/32 = 1/2.
- Thus T = 16 hours.
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