A Christmas problem

Number Theory Level 2

One of Santa's elves can make one toy each minute. Every 2 minutes 1 toy falls apart. Every 5 minutes Santa takes 3 toys. If two elves are working in a room that contains 30 toys, than how many toys will there be after an hour, assuming that Santa has just left and that the elves work for the full hour?


The answer is 84.

Jennifer D. by Jennifer D. , 40, USA

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3 solutions

Chris Lewis
Dec 8, 2020

Every ten minutes, the elves make 20 20 new toys between them; 5 5 toys break; and Santa takes 6 6 , for an overall increase of 9 9 new toys.

Over the hour, there are six ten minute periods, so there will be 30 + 6 × 9 = 84 30+6\times 9=\boxed{84} toys.

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Jennifer D.
Dec 7, 2020

Both elves working together make 2 toys each minute. In 2 minutes 4 toys are made, but one falls apart. So every 2 minutes the number of toys increases by 3. 60 minutes can be divided into 30 2-minute sections. That means that they add 90 to the 30 already there. That makes 120 all together. 60 is 12 5-minute sections, which means Santa takes 36 toys. 120 - 36 = 84, so that is the answer.

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Richard Desper
Dec 7, 2020

The room starts with 30 30 toys in it. Over the course of an hour, each of the two elves makes 60 60 new toys. Also, 30 30 toys will fall apart in one hour. Finally, Santa will visit the room 12 12 times, taking 3 3 toys each time. The final number of toys in the room is 30 + 2 ∗ 60 − 30 − 12 ∗ 3 = 120 − 36 = 84. 30 + 2*60- 30 - 12*3 = 120 - 36 = 84.

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