Problem 18 · 2002 AMC 8
Hard
Arithmetic & Operations
total-then-divide
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?
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Answer: E — 2 hours.
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Hint 1 of 2
An average is a target everyone must *balance out* to. So measure each day as how far above or below 85 it lands, not its raw minutes.
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Hint 2 of 2
Days below 85 owe minutes; days above 85 pay them back. Net the surplus against the shortfall — day 9 must cover whatever's still owed (on top of its own 85).
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Approach: measure each day against the 85-minute target
- Read every day as a *deviation* from the 85-min target. The 5 days at 75 are each 10 short → −50; the 3 days at 90 are each 5 over → +15.
- Running balance: −50 + 15 = −35, a 35-minute shortfall. Day 9 must hit the target *and* erase it: 85 + 35 = 120 min = 2 hours.
- *The transferable trick:* to hit a target average, track only the ± gaps from the target — they must cancel to zero. The last value just absorbs the leftover gap, far lighter than summing everything.
Another way — totals:
- Skated so far: 5·75 + 3·90 = 645 min. Needed for the average: 9·85 = 765 min.
- Day 9 = 765 − 645 = 120 min = 2 hours.
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