2000 AMC 10 Problems/Problem 17: Difference between revisions
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==Problem== | ==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? | 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? | ||
<math>\ | <math> \textbf{(A)} \$3.63 \qquad \textbf{(B)} \$5.13 \qquad \textbf{(C)}\$6.30 \qquad \textbf{(D)} \$7.45 \qquad \textbf{(E)} \$9.07</math> | ||
<math> | ==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 <math>124</math> cents. | |||
This implies that the only possible values, in cents, he can have are the ones one more than a multiple of <math>124</math>. Of the choices given, the only one is <math>\boxed{\text{D}}</math> | |||
==Video Solution== | |||
https://youtu.be/ZmOrAsgvS4s | |||
~savannahsolver | |||
https://youtu.be/oWxqYyW926I | |||
==Solution== | -gnv12 | ||
==Video Solution by Daily Dose of Math== | |||
https://youtu.be/4sDw2cIs_KI?si=EQyxXp4QaUnSc6N0 | |||
~Thesmartgreekmathdude | |||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2000|num-b=16|num-a=18}} | {{AMC10 box|year=2000|num-b=16|num-a=18}} | ||
{{MAA Notice}} | |||
[[Category:Introductory Number Theory Problems]] | |||
Latest revision as of 08:38, 1 May 2025
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?
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 
cents.
This implies that the only possible values, in cents, he can have are the ones one more than a multiple of . Of the choices given, the only one is 
Video Solution
~savannahsolver
-gnv12
Video Solution by Daily Dose of Math
https://youtu.be/4sDw2cIs_KI?si=EQyxXp4QaUnSc6N0
~Thesmartgreekmathdude
See Also
| 2000 AMC 10 (Problems • Answer Key • Resources) | ||
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
