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

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==Problem==
 
==Problem==
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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?
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<math>\mathrm{(A)}</math> <math>\$3.63</math>
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<math>\mathrm{(B)}</math> <math>\$5.13</math>
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<math>\mathrm{(C)}</math> <math>\$6.30</math>
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<math>\mathrm{(D)}</math> <math>\$7.45</math>
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<math>\mathrm{(E)}</math> <math>\$9.07</math>
  
 
==Solution==
 
==Solution==
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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.
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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>
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==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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