2003 AMC 12B Problems/Problem 7

Problem

Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels,dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?

$\text {(A) } 0 \qquad \text {(B) } 13 \qquad \text {(C) } 37 \qquad \text {(D) } 64 \qquad \text {(E) } 83$

Solution

Where $a,b,c$ is the number of nickels, dimes, and quarters, respectively, we can set up two equations:

$(1)\ 5a+10b+25c=835\ \ \ \ (2)\ a+b+c=100$

Eliminate $a$ by subtracting $(2)$ from $(1)/5$ to get $b+4c=67$ . Of the integer solutions $(b,c)$ to this equation, the number of dimes $b$ is least in $(3,16)$ and greatest in $(67,0)$ , yielding a difference of $67-3=\boxed{\textbf{(D)}\ 64}$ .

2003 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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All AMC 12 Problems and Solutions

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2003 AMC 12B Problems/Problem 7

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