smc

Question

Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels,dimes, and quarters, whose total value is $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?

Accepted Answer

Where a,b,c is the number of nickels, dimes, and quarters, respectively, we can set up two equations: (1) 5a+10b+25c=835 (2) a+b+c=100 Eliminate a by subtracting (2) from (1)/5 to get b+4c=67. Of the integer solutions (b,c) to this equation, the number of dimes b is least in (3,16) and greatest in (67,0), yielding a difference of 67-3=64.