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

(Solution)
(See Also)
Line 22: Line 22:
  
 
{{AMC10 box|year=2000|num-b=16|num-a=18}}
 
{{AMC10 box|year=2000|num-b=16|num-a=18}}
 +
{{MAA Notice}}

Revision as of 20:57, 3 July 2013

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?

$\mathrm{(A)}$ <dollar/>$3.63$

$\mathrm{(B)}$ <dollar/>$5.13$

$\mathrm{(C)}$ <dollar/>$6.30$

$\mathrm{(D)}$ <dollar/>$7.45$

$\mathrm{(E)}$ <dollar/>$9.07$

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 $124$ cents.

This implies that the only possible values, in cents, he can have are the ones one more than a multiple of $124$. Of the choices given, the only one is $\boxed{\text{D}}$

See Also

2000 AMC 10 (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 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. AMC logo.png

Invalid username
Login to AoPS