Problem 4 · 2016 AMC 8
Easy
Ratios, Rates & Proportions
distance-speed-timeunit-rate
When Cheenu was a boy he could run 15 miles in 3 hours and 30 minutes. As an old man he can now walk 10 miles in 4 hours. How many minutes longer does it take for him to travel a mile now compared to when he was a boy?
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Answer: B — 10 minutes longer.
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Hint 1 of 2
The question asks about ONE mile, but the two trips are different lengths — comparing the raw trip times would be unfair. Shrink each trip down to a per-mile rate so they're on equal footing.
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Hint 2 of 2
The unit that matches the question is minutes-per-mile: divide each trip's total minutes by its miles, then the answer is just the gap between the two paces.
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Approach: reduce each trip to minutes-per-mile, then subtract
- Boy: 3 h 30 min = 210 minutes for 15 miles ⇒ 210 ÷ 15 = 14 minutes per mile.
- Old man: 4 h = 240 minutes for 10 miles ⇒ 240 ÷ 10 = 24 minutes per mile.
- He's slower now, so it takes 24 − 14 = 10 extra minutes per mile.
- Sanity check: he covers fewer miles in more time as an old man, so the pace MUST be slower — a positive difference is expected, and 10 is small enough to be one mile's worth (not the whole trip).
- You'll see this again as: any rate comparison — convert both to the SAME unit the question asks about (here, minutes per mile) before comparing; never compare totals of different sizes.
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