AMC 10 · 2012 · #2

Grade 5 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
rateunit-conversion dimensional-analysisidentify-subproblems ↑ Prerequisites: ratemulti-digit-arithmetic
📏 Medium solution 💡 1 insight
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Problem
Two people work at their own steady rates at the same time. Find how many they finish together.

Pick an answer.

(A)
10
(B)
15
(C)
20
(D)
25
(E)
30
How to solve
Strategy Analyze the Units

The rates are stated in seconds but the working time is in minutes, so Tool #8 (Analyze the Units) is the key move: convert 5 minutes to 300 seconds so every quantity is measured the same way. Once the units line up, Tool #7 (Identify Subproblems) splits the job into two easy pieces — count Cagney's cupcakes, count Lacey's cupcakes — and then adds them, since they work at the same time.

1STEP 1

Match the units

The work time becomes 300 seconds.

5 min = 5 × 60 = 300 s
2STEP 2

Count Cagney's cupcakes

The faster one finishes 15.

300 ÷ 20 = 15
3STEP 3

Count Lacey's cupcakes

The slower one finishes 10.

300 ÷ 30 = 10
4STEP 4

Add the two counts

Adding gives 25, choice (D).

15 + 10 = 25 → (D)
Answer
25
Cagney is the faster froster, so she should make more than Lacey — and 15 is more than 10, which fits. A rough estimate agrees too: their combined speed is a bit faster than one cupcake every 12 seconds, and 300 seconds divided by 12 is 25, matching the answer. The trap choices come from stopping early: 15 is Cagney alone (B) and 10 is Lacey alone (A); the question asks for both, so 25 is correct.
💡Key takeaway

Make the units match first, then divide the total time by each person's time-per-cupcake and add the counts.

  • Match the units
  • Count Cagney's cupcakes
  • Count Lacey's cupcakes
  • Add the two counts

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