AMC 10 · 2024 · #8
Grade 6 rate-ratioalgebraProblem
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at 1:00 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 PM. At what time did Daria join the group?
Pick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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
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
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.
💡 Rates with the same time unit add directly — a Grade 6 unit-rate move.
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.
💡 Once the rate sits at "$10$ per $3$ min," total work is just (number of chunks) $\times$ (packages per chunk).
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.
💡 The total acts as a constraint — subtract the known piece to reveal the unknown piece.
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.
💡 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.
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.
💡 Subtracting an elapsed time from an end time is a Grade 4 measurement skill — the same logic as a number line in reverse.
6.RP.A.2 Subproblem 1: combine the three steady rates. Amy, Bomani, and Charlie all repor 6.RP.A.3 Subproblem 2: count how many packages the first three pack across the entire shi 4.OA.A.3 Subproblem 3: use the $450$-package total to isolate Daria's contribution. Whate 6.RP.A.3 Subproblem 4: convert Daria's package count into time using her rate. Daria pack 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.2Understand 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.3Use 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.3Solve 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.2Use 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.
⭐ 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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