AMC 10 · 2012 · #1
Grade 5 rate-ratioPick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Match the units
The rates are per second but the time is 5 minutes, so convert first: 5 × 60 = 300 seconds.
You can only compare or combine quantities when they are measured in the same unit.
4.MD.A.1Analyze The UnitsCount Cagney's cupcakes
Cagney takes 20 seconds each, so she frosts 300 ÷ 20 = 15 cupcakes.
Dividing the total time by the time-per-cupcake counts how many cupcakes fit in that time.
Dividing the total time by the time each one takes counts how many fit into that stretch.
▸ Why?
At a steady pace the amount done is the rate times the time, so dividing recovers the count.
▸ Why?
Each one takes the same fixed stretch of time, so the stretches stack without gaps.
Count Lacey's cupcakes
Lacey takes 30 seconds each, so she frosts 300 ÷ 30 = 10 cupcakes.
The slower worker fits fewer cupcakes into the same amount of time.
5.NBT.B.6Identify SubproblemsAdd the two counts
They frost at the same time, so the counts simply add: 15 + 10 = 25 cupcakes, choice (D).
When two people work during the same stretch of time, their separate totals just add up.
4.NBT.B.4Identify SubproblemsMake 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