AMC 8 · 2001 · #15

Grade 6 rate-ratioalgebra

Archetype Meeting Time with Piecewise Rates └ Sub-archetype Staggered Start or Schedule Table

ratelinear-equations-one-varmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: ratemulti-digit-arithmetic

📏 Medium solution 💡 3 insights

Problem

Homer starts peeling a pile of 44 potatoes at 3 per minute. After 4 minutes Christen joins and peels at 5 per minute. They keep peeling until the pile is gone. How many potatoes did Christen peel?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The story has two distinct phases — Homer alone for the first 4 minutes, then Homer and Christen together. Tool #7 (Identify Subproblems) splits the work along that natural seam: (a) count what Homer peeled alone and subtract from 44 to get what is left, then (b) work out how long the two of them need to finish that remainder, and finally read off Christen's share. Tool #8 (Analyze the Units) keeps every step honest: every multiplication is potatoes/minute × minutes = potatoes, and the rate-addition step pools 3 + 5 correctly because both rates share the same unit.

1STEP 1

Count Homer's solo work: 3 per minute for his 4 minutes gives 12 potatoes.

3 potatoes/minute × 4 minutes = 12 potatoes

2STEP 2

Subtract from the pile: 44 minus 12 leaves 32 potatoes for the joint phase.

44 - 12 = 32 potatoes remaining

3STEP 3

They share one pile, so add the rates: 3 plus 5 makes 8 potatoes per minute.

3 + 5 = 8 potatoes/minute

4STEP 4

Divide the remainder by the joint rate: 32 over 8 gives 4 minutes of shared peeling.

(32 potatoes)/(8 potatoes/minute) = 4 minutes

5STEP 5

Christen worked only that shared span: 5 per minute for 4 minutes is 20 potatoes.

5 potatoes/minute × 4 minutes = 20 potatoes → (A)

Answer

Cross-check the totals. Homer peeled 12 alone plus 3 × 4 = 12 more during phase 2, so Homer peeled 24. Christen peeled 20. Together that is 24 + 20 = 44 potatoes — the whole pile, exactly as required. The trap answers fit common slips. (C) 32 is the leftover pile, not Christen's share. (B) 24 is Homer's total. (D) 33 matches splitting the original 44 by the rate ratio 5:3 as if Christen had peeled the whole time. (E) 40 comes from giving Christen all 8 minutes at 5/minute, ignoring her 4-minute late start. Choosing (A) means the late start was handled correctly.

💡Key takeaway

Split the story into two phases — Homer alone, then both together. Subtract what Homer peeled alone, add the rates for the joint phase, and Christen's 4 minutes at 5/minute give 20 potatoes, answer (A).

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