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

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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>\mathrm{(A)}</math> <dollar/><math>3.63</math>  
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<math>\mathrm{(A)}</math> <math>\$3.63</math>  
  
<math>\mathrm{(B)}</math> <dollar/><math>5.13</math>
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<math>\mathrm{(B)}</math> <math>\$5.13</math>
  
<math>\mathrm{(C)}</math> <dollar/><math>6.30</math>  
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<math>\mathrm{(C)}</math> <math>\$6.30</math>  
  
<math>\mathrm{(D)}</math> <dollar/><math>7.45</math>  
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<math>\mathrm{(D)}</math> <math>\$7.45</math>  
  
<math>\mathrm{(E)}</math> <dollar/><math>9.07</math>
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<math>\mathrm{(E)}</math> <math>\$9.07</math>
  
 
==Solution==
 
==Solution==
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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>
 
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 by WhyMath==
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https://youtu.be/ZmOrAsgvS4s
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 +
~savannahsolver
  
 
==See Also==
 
==See Also==
  
 
{{AMC10 box|year=2000|num-b=16|num-a=18}}
 
{{AMC10 box|year=2000|num-b=16|num-a=18}}
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{{MAA Notice}}

Revision as of 22:16, 23 February 2021

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 $124$ 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 $\boxed{\text{D}}$

Video Solution by WhyMath

https://youtu.be/ZmOrAsgvS4s

~savannahsolver

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

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