Problem:
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Four pennies are needed to make small change. Adding two nickels means that change up to fourteen cents can be made as efficiently as possible. Adding a dime extends the efficiency up to cents. Adding three quarters permits any amount of change up to .
Note that pennies, nickel, dimes, and quarters also satisfies the requirements. For to be the smallest number of coins, one must prove that coins will not work. Freddie will still need at least pennies and nickel to make any amount up to . Freddie will still need at least more nickel and dime to make any amount up to . If Freddie chose only quarters, then he would need at least other coins dimes, nickel, and pennies) to reach cents. He would need at least coins. Thus is the smallest number of coins as shown above.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions