AMC 10 · 2012 · #2
Easy mode Grade 5Cagney frosts one cupcake every 20 seconds. Lacey frosts one cupcake every 30 seconds. They both work at the same time for 5 minutes. How many cupcakes do they frost in all?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Cagney frosts one cupcake every 20 seconds and Lacey frosts one every 30 seconds. Working at the same time, find how many cupcakes they finish together in 5 minutes.
Givens: Cagney's rate: one cupcake per 20 seconds; Lacey's rate: one cupcake per 30 seconds; Total working time: 5 minutes; Answer choices: (A) 10, (B) 15, (C) 20, (D) 25, (E) 30
Unknowns: The total number of cupcakes both people frost together in 5 minutes
Understand
Restated: Cagney frosts one cupcake every 20 seconds and Lacey frosts one every 30 seconds. Working at the same time, find how many cupcakes they finish together in 5 minutes.
Givens: Cagney's rate: one cupcake per 20 seconds; Lacey's rate: one cupcake per 30 seconds; Total working time: 5 minutes; Answer choices: (A) 10, (B) 15, (C) 20, (D) 25, (E) 30
Plan
Primary tool: #8 Analyze the Units
Secondary: #7 Identify Subproblems
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.
Execute — Answer: D
4.MD.A.1 Step 1 Match the units
- The rates are per second but the time is 5 minutes, so change minutes into seconds before doing anything else.
- One minute is 60 seconds, so 5 minutes is $5 \times 60 = 300$ seconds.
- Now the total time and both rates all use seconds.
💡 You can only compare or combine quantities when they are measured in the same unit.
5.NBT.B.6 Step 2 Count Cagney's cupcakes
- Cagney finishes one cupcake every 20 seconds, so the number she frosts is how many 20-second blocks fit into 300 seconds.
- Divide: $300 \div 20 = 15$.
- Cagney frosts 15 cupcakes.
💡 Dividing the total time by the time-per-cupcake counts how many cupcakes fit in that time.
5.NBT.B.6 Step 3 Count Lacey's cupcakes
- Lacey finishes one cupcake every 30 seconds, so divide the same 300 seconds by 30: $300 \div 30 = 10$.
- Lacey frosts 10 cupcakes.
💡 The slower worker fits fewer cupcakes into the same amount of time.
4.NBT.B.4 Step 4 Add the two counts
- They frost at the same time, so the cupcakes they finish simply add together: $15 + 10 = 25$.
- Together they frost 25 cupcakes, which is choice (D).
💡 When two people work during the same stretch of time, their separate totals just add up.
4.MD.A.1 The rates are per second but the time is 5 minutes, so change minutes into secon 5.NBT.B.6 Cagney finishes one cupcake every 20 seconds, so the number she frosts is how ma 5.NBT.B.6 Lacey finishes one cupcake every 30 seconds, so divide the same 300 seconds by 3 4.NBT.B.4 They frost at the same time, so the cupcakes they finish simply add together: $1 Review
Reasonableness: 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.
Alternative: Work in cupcakes-per-minute instead. Cagney frosts 3 per minute (one every 20 s) and Lacey frosts 2 per minute (one every 30 s), so together they frost 5 per minute. Over 5 minutes that is $5 \times 5 = 25$ cupcakes, confirming (D).
CCSS standards used (min grade 5)
4.MD.A.1Know relative sizes of measurement units and convert larger to smaller units (Converting 5 minutes into 300 seconds so the time matches the per-second rates.)5.NBT.B.6Find whole-number quotients with up to four-digit dividends and two-digit divisors (Dividing 300 by the two-digit divisors 20 and 30 to count each person's cupcakes.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Adding Cagney's 15 and Lacey's 10 to get the combined total of 25.)
⭐ Make the units match first, then divide the total time by each person's time-per-cupcake and add the counts.
⭐ Make the units match first, then divide the total time by each person's time-per-cupcake and add the counts.
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