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2003 AMC 12B Problem 7

Problem 7 of 25EasierAlgebraNumber Theory

Penniless Pete’s piggy bank has no pennies in it, but it has 100100 coins, all nickels, dimes, and quarters, whose total value is $8.35.\$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?

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Solution

Let n,n, d,d, qq be the numbers of nickels, dimes, quarters. Then n+d+q=100n + d + q = 100 and n+2d+5q=167n + 2d + 5q = 167 (dividing the value equation by 55). Subtracting gives d+4q=67,d + 4q = 67, so d=67−4q.d = 67 - 4q. The largest dd is at q=0,q = 0, giving d=67d = 67 (with n=33n = 33). The smallest occurs at q=16,q = 16, giving d=3d = 3 (with n=81n = 81). The difference is 67−3=64.67 - 3 = 64. Thus, the correct answer is D.

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Concepts: system of equations · Diophantine Equation · bounding to limit cases

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.