Problem:
Pablo buys popsicles for his friends. The store sells single popsicles for each, -popsicle boxes for , and -popsicle boxes for . What is the greatest number of popsicles that Pablo can buy with ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The cheapest popsicles cost each. Because and Pablo has just , he could not pay for popsicles even if he were allowed to buy partial boxes. The best he can hope for is popsicles, and he can achieve that by buying two -popsicle boxes (for ) and one -popsicle box (for ).
If Pablo buys two single popsicles for each, he could have bought a -popsicle box for the same amount of money. Similarly, if Pablo buys three single popsicles or both one -popsicle box and one single popsicle, he could have bought a -popsicle box for the same amount of money. If Pablo buys two -popsicle boxes, he could have bought a -popsicle box and a single popsicle for the same amount of money. The previous statements imply that a maximum number of popsicles for a given amount of money can be obtained by buying either at most one single popsicle and the rest -popsicle boxes, or a single -popsicle box and the rest -popsicle boxes. When Pablo has , he can obtain the maximum number of popsicles by buying two -popsicle boxes and one -popsicle box. This gives a total of popsicles.
The problems on this page are the property of the MAA's American Mathematics Competitions