2017 AMC 10A Problem 4

The count goes up by $3$ and then down by $2$, so the box is fullest right after Mom's drop. That is the moment to watch.

1. Write down what happens in the first few half-minutes:

1. The pattern: right after Mom's drop in half-minute number $k$, the box has $k + 2$ toys. (Each round nets one toy, and Mom's drop puts $2$ extra in on top for a moment.)
2. We want the first time the box holds $30$ toys. Set $k + 2 = 30$, so $k = 28$.
3. At that instant all $30$ toys are in the box. What Mia does afterward does not matter, because the question asks for the first time.
4. Convert: $28$ half-minutes is $28 \times 30 = 840$ seconds, which is $840 \div 60 = 14$ minutes.

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

A "net rate" of $1$ toy per round tells you the long-run trend, but the goal can be reached in the middle of a round. In any up-and-down process, check the high point of each round, not just the end of it. A small table of the first few rounds shows the pattern clearly.

Count full rounds first. After $27$ full rounds (Mom adds $3$, Mia removes $2$) there are $27$ toys in the box. In round $28$, Mom adds $3$ and the box holds $30$. That is $28$ half-minutes, or $14$ minutes.