AMC 10 · 2017 · #4
Grade 5 rate-ratioPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem is a rate-of-progress question — toys per 30 seconds — so Tool #8 (Analyze the Units) anchors it: track the net toys added each 30-second block. Tool #5 (Look for a Pattern) turns that net change into a repeating +1 step we can multiply. The trap is the very last block, where the job finishes before Mia can remove anything, so Tool #14 (Extreme Principle) forces us to inspect that boundary block separately instead of applying the pattern blindly.
Net toys per 30 seconds
Each 30-second cycle Mom adds 3 and Mia removes 2, so the box nets 1 toy every 30 seconds.
Adding then removing in the same block is one combined step, so only the leftover +1 actually counts.
4.OA.A.3Analyze The UnitsThe last push is different
Mia removes only after a block ends, so Mom's +3 that first reaches 30 ends the job — the net +1 pattern only has to climb to 27.
The clock stops the instant the box is full, so the last handful of toys never has a chance to come back out.
4.OA.A.3Evaluate Finite DifferencesCount the blocks of time
27 blocks at +1 take 27 × 30 = 810 s, then one more 30-s block adds Mom's final +3, for a total of 840 seconds.
Each net toy costs exactly one 30-second block, plus one extra block for the closing push.
4.OA.A.3Look For A PatternConvert seconds to minutes
A minute is 60 seconds, so divide: 840 ÷ 60 = 14 minutes, choice (B).
Seconds and minutes measure the same time, so dividing by 60 just renames the amount.
5.MD.A.1Analyze The UnitsWhen something is added then taken away each round, track the net change — but check the final round, because the job can finish before the take-away happens.
- Net toys per 30 seconds
- The last push is different
- Count the blocks of time
- Convert seconds to minutes