Bunuel wrote:
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? (Dime= 10 cents; Nickel= 5 cents; Penny= 1 cent)
(A) 0
(B) 13
(C) 37
(D) 64
(E) 83
Let n, d, and q be the number of nickels, dimes, and quarters, respectively. We can generate two equations:
n + d + q = 100 (Number of coins equation)
and
5n + 10d + 25q = 835 (Money value equation, in cents)
n + 2d + 5q = 167
Subtracting the first equation from the second, we have:
d + 4q = 67
To maximize the number of dimes, we minimize the number of quarters. If we have no quarters (q = 0), we will have 67 dimes, and this is the maximum number of dimes.
Now, to minimize the number of dimes, we must maximize the number of quarters. So 4q must be as large as possible, and this occurs when q = 16. When q = 16, then d = 3. Thus, the minimum number of dimes is 3.
The difference between the maximum and minimum number of dimes is 67 - 3 = 64.
Answer: D
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