AMC 10 · 2024 · #8

Grade 6 rate-ratioalgebra

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

rateunit-conversionlinear-equations-one-var identify-subproblemsdimensional-analysis ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Medium solution 💡 3 insights

Problem

Amy, Bomani, and Charlie start packing at 1/:00 PM with rates of 4, 3, and 3 packages every 3 minutes. Daria joins later, packing 5 packages every 4 minutes. The group finishes 450 packages at 2/:45 PM. At what time did Daria join?

Pick an answer.

(A)

$1:25\text{ PM}$

(B)

$1:35\text{ PM}$

(C)

$1:45\text{ PM}$

(D)

$1:55\text{ PM}$

(E)

$2:05\text{ PM}$

AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question hides two separate jobs: how much the first three workers pack across the whole shift, and how much Daria has to add on top to hit 450. Tool #7 (Identify Subproblems) splits the work cleanly along those lines — combine the three rates, multiply by the full 105 minutes, then subtract from 450 to isolate Daria's contribution. Once we know how many packages Daria did, her rate tells us her working time. From there, Tool #11 (Work Backwards) is the natural finish: we know when she stopped (2{:}45 PM) and how long she worked, so we step backward to her start time.

1STEP 1

Same 3-minute window for all three, so add the rates: together they pack 10 packages every 3 minutes.

A+B+C rate = 4 + 3 + 3 = 10 packages per 3 minutes

2STEP 2

Across the full 105-minute shift, 35 three-minute chunks of 10 give the three a total of 350 packages.

35 × 10 = 350 packages by A, B, C

3STEP 3

The 450 total minus the three's 350 leaves Daria with 100 packages.

Daria's packages = 450 - 350 = 100

4STEP 4

At 5 packages per 4 minutes, each takes $\frac{4}{5}$ minute, so her 100 packages fill 80 minutes.

Daria's time = 100 ×

\frac{4}{5}

= 80 minutes

5STEP 5

She stopped at 2/:45 PM after 80 minutes, so rewind 1 hour 20 minutes to reach 1/:25 PM.

2{:}45 PM - 1h 20min = 1{:}45 PM - 20 min = 1{:}25 PM → (A)

Answer

1:25 PM

Double-check by going forward from 1/:25 PM. Amy, Bomani, Charlie pack for 105 minutes at 10 per 3 min = 350 packages. Daria works from 1/:25 PM to 2/:45 PM, which is 80 minutes, at 5 per 4 min =

\frac{80}{4}

× 5 = 20 × 5 = 100 packages. Total: 350 + 100 = 450. Matches the problem exactly, so 1/:25 PM is confirmed.

💡Key takeaway

Split the job — three workers cover the whole shift, Daria fills the gap — then rewind 80 minutes from 2/:45 PM and the AMC 10 problem lands on a Grade 6 rate-and-clock question.

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