Problem 15 · 2001 AMC 8
Hard
Ratios, Rates & Proportions
work-rateunit-rate
Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled?
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Answer: A — 20.
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Hint 1 of 2
Christen's count depends on how LONG she works, so the real target is the shared time — peel away Homer's head start first.
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Hint 2 of 2
After the head start, the two work together at a combined rate; that combined rate tells you how long the rest takes.
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Approach: head start, then combined rate
- Account for the head start: in 4 minutes alone Homer peels 3 × 4 = 12, leaving 44 − 12 = 32 potatoes.
- Now they work side by side at 3 + 5 = 8 potatoes per minute, so finishing 32 takes 32 ÷ 8 = 4 minutes. (Combine rates by adding when people work at the same time.)
- Christen worked all 4 of those minutes at 5 per minute: 5 × 4 = 20 potatoes. The key was finding the shared time first — counts follow once you know how long someone worked.
Another way — split the remaining work by rate share:
- After the head start, 32 potatoes remain and the pair peels in a 3 : 5 ratio (Homer : Christen), 8 parts total.
- Christen does 5/8 of the remaining 32 = 20 potatoes — no need to find the time at all.
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