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2017 AMC 10A Problem 4

Problem 4 of 25EasierAlgebra

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?

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Solution

Note that after 3030 seconds, there are 33 toys added and 22 removed, leaving a net total of +1+1 toys in the box. We have to be careful towards the end, however, since it is possible for the box to have 3030 toys right after Mia’s mom adds the toys and before Mia removes them. After there are 2727 toys in the box, Mia’s mom can add 3,3, leaving 3030 toys in the box. It will take 273027 \cdot 30 seconds, plus another 3030 seconds, which gives us 2830÷60=14 28 \cdot 30 \div 60 = 14 minutes to get 3030 toys in the box. Thus, B is the correct answer.

More practice

Concepts: rate · process simulation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.