# Difference between revisions of "2018 AMC 10A Problems/Problem 8"

Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins? $\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$

## Solution

Let $x$ be the number of 5-cent stamps that Joe has. Therefore, he must have $(x+3)$ 10-cent stamps and $(23-(x+3)-x)$ 25-cent stamps. Since the total value of his collection is 320 cents, we can write

\begin{align*} 5x+10(x+3)+25(23-(x+3)-x)= 320 \\ &\Rightarrow 5x+10(x+3)+25(20-2x)= 320 \\ &\Rightarrow 5x+10x+30+500-50x= 320 \\ &\Rightarrow 35x= 210 \\ &\Rightarrow x= 6 \ \end{align*}

Joe has 6 5-cent stamps, 9 10-cent stamps, and 8 25-cent stamps. Thus, our answer is $8-6=\boxed{2}$

~Nivek

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