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2017 AMC 10A Problem 4

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2017 AMC 10A Problem 4AMC 10 Step by Step
Name:                             Date: September 13, 2026
Games & Processes · difficulty 2 of 5 · about 2 min · position P1-10

Mia is “helping” her mom pick up 3030 toys that are strewn on the floor. Mia’s mom manages to put 33 toys into the toy box every 3030 seconds, but each time immediately after those 3030 seconds have elapsed, Mia takes 22 toys out of the box. How much time, in minutes, will it take Mia and her mom to put all 3030 toys into the box for the first time?

  1. A)

    13.513.5

  2. B)

    1414

  3. C)

    14.514.5

  4. D)

    1515

  5. E)

    15.515.5

Answer

B

Key insight

Each 30-second cycle nets one toy, but the box reaches 30 the instant Mom drops in her third batch, before Mia can react, so 27 cycles plus one more deposit.

Solution

Think of each 3030-second period as a cycle: Mom adds 33 toys at the end of the period, and Mia immediately removes 22. The net gain per cycle is 11 toy, so after kk complete cycles (Mom's deposit followed by Mia's removal) there are kk toys in the box.

The count reaches 3030 at the moment Mom deposits, not after Mia removes. After 2727 complete cycles there are 2727 toys. During the 2828th period Mom adds 33 more, and at that instant the box holds 3030 toys for the first time. Mia's removal afterward is irrelevant.

That moment is 2828 periods of 3030 seconds: 28⋅30=84028 \cdot 30 = 840 seconds =14= 14 minutes.

The answer is (B) 14\boxed{\textbf{(B)}\ 14}.

Why this works

"Net rate" reasoning gives the right long-run picture but can miss the finish line, because the target is hit in the middle of a cycle. In any back-and-forth process, always ask what the count looks like at the peak of a cycle, not just at the end of one. Here the peak after kk deposits is k+2k + 2, and k+2=30k + 2 = 30 at k=28k = 28.

Alternative approach

Track peaks: after Mom's first deposit there are 33, then 44 after the second, then 55, and so on: the kkth deposit brings the box to k+2k + 2 toys. Setting k+2=30k + 2 = 30 gives k=28k = 28 deposits, i.e. 2828 half-minutes, which is 1414 minutes.

The trap

Dividing 30 toys by the net rate of 1 toy per half-minute and answering 15 minutes.

Common mistakes

  • Dividing 30 toys by the net rate of 1 toy per half-minute and answering 15 minutes.
  • Using 2727 or 2929 cycles from an off-by-one error and choosing 13.513.5 or 14.514.5.

Techniques

Set up the equation/formula and compute; no special trick needed · Compute small cases, spot the pattern, generalize