Competition · AMC preparation · step 4 of 4
AMC 8 · 2001 · #15
Grade 6 rate-ratioalgebraPick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Count Homer's first 4 minutes
Count Homer's solo work: 3 per minute for his 4 minutes gives 12 potatoes.
Rate × time = amount. The minute units cancel, leaving potatoes — exactly what we want to count.
5.NBT.B.5Analyze The UnitsFind the potatoes left
Subtract from the pile: 44 minus 12 leaves 32 potatoes for the joint phase.
Total minus what is done leaves what is left — the bridge between phase 1 and phase 2.
4.OA.A.3Identify SubproblemsAdd the two rates
They share one pile, so add the rates: 3 plus 5 makes 8 potatoes per minute.
Rates with the same units add directly. Two peelers acting together is one combined rate of 8 potatoes per minute.
5.NBT.B.5Analyze The UnitsFind the shared work time
Divide the remainder by the joint rate: 32 over 8 gives 4 minutes of shared peeling.
Dividing potatoes by potatoes-per-minute leaves minutes — the time to finish at the joint pace.
6.RP.A.3Analyze The UnitsCount Christen's potatoes
Christen worked only that shared span: 5 per minute for 4 minutes is 20 potatoes.
Christen only worked during phase 2, so her share is her rate times the phase-2 duration — nothing more, nothing less.
Christen peeled a number of potatoes equal to her steady peeling rate multiplied by the length of phase two — the only stretch of time she was actually peeling.
▸ Why?
Peeling at a steady rate, the potatoes someone finishes equal that rate multiplied by the minutes spent — rate times time gives an amount.
▸ Why?
A steady rate of five potatoes a minute means five potatoes in every single minute, so the total across the minutes is just that many equal groups of five added together.
▸ Why?
The only minutes Christen was peeling belong to phase two; she joined after the pile was already shrinking, so her share during phase one was nothing at all.
▸ Why?
Her phase-one count is zero, and adding that zero to her phase-two count leaves the phase-two count unchanged, so phase two alone decides her total.
▸ Why?
Phase two runs for a definite number of minutes: the potatoes still in the pile when Christen joins, cleared at the two peelers' combined rate.
▸ Why?
The time to clear a fixed pile at a steady rate is the pile size divided by that rate, because dividing undoes the rate-times-time multiplication that measured the work.
▸ Why?
The combined rate is the two separate rates added, because in each minute the pile loses Homer's potatoes and Christen's potatoes with no overlap, and that minute's whole loss is the two amounts joined.
▸ Why?
The potatoes still in the pile when Christen joins are the whole starting pile with Homer's solo-peeled potatoes taken away, since the pile splits with no overlap into what is already done and what is left.
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).
- Count Homer's first 4 minutes
- Find the potatoes left
- Add the two rates
- Find the shared work time
- Count Christen's potatoes
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