Question

Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive 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?

$62$

$82$

$83$

$102$

$103$

Solution:

After $m$ red-booth and $n$ blue-booth exchanges, Alex has $75 - (2m - n)$ red tokens, $75 - (3n - m)$ blue tokens, and $m + n$ silver tokens. Exchanges are impossible exactly when $2m - n \ge 74$ and $3n - m \ge 73.$ Equality holds at $(m, n) = (59, 44),$ giving $59 + 44 = 103$ silver tokens. Thus, the correct answer is E.

2013 AMC 12B Problem 10

Solution:

Problem 10 in Other Years