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 . 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?
Answer Choices:
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E.
Solution:
Let represent the number of nickels in the bank, the number of dimes, and the number of quarters. Then and . Further, , , and must all be nonnegative integers. Dividing the second equation by 5 yields . Subtracting the first equation from this gives . Since cannot be negative, is at most 67 , and we check that
67 dimes and 33 nickels indeed produces . On the other hand, cannot be 0,1 , or 2 because then would not be an integer. Thus the smallest can be is 3 , leaving . We check that 16 quarters, 3 dimes, and 81 nickels also produces . Thus the largest can be is 67 , the smallest is 3 , and the difference is .
The problems on this page are the property of the MAA's American Mathematics Competitions