AMC 10 · 2024 · #8

Grade 6 rate-ratioalgebra

Archetype Meeting Time with Piecewise Rates

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

📏 Medium solution 💡 3 insights

Problem

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1:00 \ \mathrm{PM}$ and were able to pack $4$ , $3$ , and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2:45\ \mathrm{PM}$ . At what time did Daria join the group?

$\textbf{(A) }1:25\text{ PM}\qquad\textbf{(B) }1:35\text{ PM}\qquad\textbf{(C) }1:45\text{ PM}\qquad\textbf{(D) }1:55\text{ PM}\qquad\textbf{(E) }2:05\text{ PM}$

Pick an answer.

(A)

$1:25 ext{ PM}$

(B)

$1:35 ext{ PM}$

(C)

$1:45 ext{ PM}$

(D)

$1:55 ext{ PM}$

(E)

$2:05 ext{ PM}$

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Amy, Bomani, Charlie start at $1{:}00$ PM and pack $4$, $3$, $3$ packages per $3$ minutes; Daria's rate is $5$ packages per $4$ minutes; Group finishes $450$ packages at $2{:}45$ PM; Answer choices: (A) $1{:}25$ PM, (B) $1{:}35$ PM, (C) $1{:}45$ PM, (D) $1{:}55$ PM, (E) $2{:}05$ PM

Unknowns: The time Daria joined the group

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #11 Work Backwards

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.

Execute — Answer: A

#7 Identify Subproblems 6.RP.A.2 Step 1

Subproblem 1: combine the three steady rates.
Amy, Bomani, and Charlie all report their rates over the same $3$-minute window, so we can simply add the package counts.

$$\text{A+B+C rate} = 4 + 3 + 3 = 10 \text{ packages per } 3 \text{ minutes}$$

💡 Rates with the same time unit add directly — a Grade 6 unit-rate move.

#7 Identify Subproblems 6.RP.A.3 Step 2

Subproblem 2: count how many packages the first three pack across the entire shift.
The full work time is $2{:}45$ PM $- 1{:}00$ PM $= 1$ hour $45$ minutes $= 105$ minutes.
Split it into $3$-minute chunks: $105 \div 3 = 35$ chunks, each producing $10$ packages.

$$35 \times 10 = 350 \text{ packages by A, B, C}$$

💡 Once the rate sits at "$10$ per $3$ min," total work is just (number of chunks) $\times$ (packages per chunk).

#7 Identify Subproblems 4.OA.A.3 Step 3

Subproblem 3: use the $450$-package total to isolate Daria's contribution.
Whatever the first three did not pack, Daria must have packed.

$$\text{Daria's packages} = 450 - 350 = 100$$

💡 The total acts as a constraint — subtract the known piece to reveal the unknown piece.

#7 Identify Subproblems 6.RP.A.3 Step 4

Subproblem 4: convert Daria's package count into time using her rate.
Daria packs $5$ packages per $4$ minutes, so each package takes $\tfrac{4}{5}$ of a minute.

$$\text{Daria's time} = 100 \times \dfrac{4}{5} = 80 \text{ minutes}$$

💡 Multiplying by the reciprocal of the rate converts "how many packages" into "how many minutes" — the bread and butter of Grade 6 unit-rate work.

#11 Work Backwards 4.MD.A.2 Step 5

Work backwards from the finish to find Daria's start.
She stopped at $2{:}45$ PM and worked for $80$ minutes ($1$ hour $20$ minutes), so step that interval backward on the clock.

$$2{:}45 \text{ PM} - 1\text{h }20\text{min} = 1{:}45 \text{ PM} - 20 \text{ min} = 1{:}25 \text{ PM} \;\Rightarrow\; \textbf{(A)}$$

💡 Subtracting an elapsed time from an end time is a Grade 4 measurement skill — the same logic as a number line in reverse.

[1] #7 6.RP.A.2 Subproblem 1: combine the three steady rates. Amy, Bomani, and Charlie all repor

[2] #7 6.RP.A.3 Subproblem 2: count how many packages the first three pack across the entire shi

[3] #7 4.OA.A.3 Subproblem 3: use the $450$-package total to isolate Daria's contribution. Whate

[4] #7 6.RP.A.3 Subproblem 4: convert Daria's package count into time using her rate. Daria pack

[5] #11 4.MD.A.2 Work backwards from the finish to find Daria's start. She stopped at $2{:}45$ PM

Review

Reasonableness: 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 $= \tfrac{80}{4} \times 5 = 20 \times 5 = 100$ packages. Total: $350 + 100 = 450$. Matches the problem exactly, so $1{:}25$ PM is confirmed.

Alternative: Tool #3 (Eliminate Possibilities) on the answer choices: for each choice, compute Daria's working minutes (end time $-$ start time), translate into packages at $5$ per $4$ min, add the fixed $350$ packages from A+B+C, and keep only the choice that hits $450$. Choice (A) $1{:}25$ PM gives $80$ min $\Rightarrow 100$ packages $\Rightarrow 450$ total. Every other choice falls short, so (A) is the answer. This trades the cleaner algebra for a direct walk through the menu.

CCSS standards used (min grade 6)

6.RP.A.2 Understand the concept of a unit rate $a/b$ associated with a ratio $a{:}b$ (Reading each worker's "packages per $3$ minutes" as a unit-rate-style ratio and combining the three rates over the shared $3$-minute window.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Scaling the combined rate $10$ packages per $3$ minutes up to $105$ minutes, and converting Daria's $100$ packages back into $80$ minutes via her $5$-per-$4$-minute rate.)
4.OA.A.3 Solve multistep word problems with whole numbers and the four operations (Using the total of $450$ packages as a constraint to subtract the $350$ done by A, B, C and isolate Daria's $100$ packages.)
4.MD.A.2 Use the four operations to solve word problems involving distances, time intervals, and money (Subtracting Daria's $80$-minute working span from the $2{:}45$ PM finish time to find her $1{:}25$ PM start time.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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