2013 AMC 10B Problems/Problem 17

Problem

Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and recieve in return a silver token and a blue token, and another booth where Alex can give three blue tokens and recieve in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

$\textbf{(A)} 62 \qquad \textbf{(B)} 82 \qquad \textbf{(C)} 83 \qquad \textbf{(D)} 102 \qquad \textbf{(E)} 103$

Solution

We can approach this problem by assuming he goes to the red booth first. You start with $75 \text{R}$ and $75 \text{B}$ and at the end of the first booth, you will have $1 \text{R}$ and $112 \text{B}$ and $37 \text{S}$ . We now move to the blue booth, and working through each booth until we have none left, we will end up with: $1 \text{R}$ , $2 \text{B}$ and $103 \text{S}$ .

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2013 AMC 10B Problems/Problem 17

Problem

Solution