Question

Mia is "helping" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?

$13.5$

$14$

$14.5$

$15$

$15.5$

Solution:

Note that after $30$ seconds, there are $3$ toys added and $2$ removed, leaving a net total of $+1$ toys in the box.

We have to be careful towards the end, however, since it is possible for the box to have $30$ toys right after Mia's mom adds the toys and before Mia removes them.

After there are $27$ toys in the box, Mia's mom can add $3,$ leaving $30$ toys in the box.

It will take $27 \cdot 30$ seconds, plus another $30$ seconds, which gives us $28 \cdot 30 \div 60 = 14$ minutes to get $30$ toys in the box.

Thus, B is the correct answer.

2017 AMC 10A Problem 4

Solution:

Problem 4 in Other Years