# Difference between revisions of "2000 AMC 10 Problems/Problem 17"

## Problem

Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?

$\mathrm{(A)}$ $3.63$

$\mathrm{(B)}$ $5.13$

$\mathrm{(C)}$ $6.30$

$\mathrm{(D)}$ $7.45$

$\mathrm{(E)}$ $9.07$

## Solution

Consider what happens each time he puts a coin in. If he puts in a quarter, he gets five nickels back, so the amount of money he has doesn't change. Similarly, if he puts a nickel in the machine, he gets five pennies back and the money value doesn't change. However, if he puts a penny in, he gets five quarters back, increasing the amount of money he has by $24$ cents.

This implies that the only possible values, in cents, he can have are the ones one more than a multiple of $24$. Of the choices given, the only one is $\boxed{\text{D}}$

## See Also

 2000 AMC 10 (Problems • Answer Key • Resources) Preceded byProblem 16 Followed byProblem 18 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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