Problem:
Pablo buys popsicles for his friends. The store sells single popsicles for each, 3-popsicle boxes for , and 5 -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 14 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 5-popsicle boxes (for $6) and one 3-popsicle box (for ).
OR
If Pablo buys two single popsicles for each, he could have bought a 3-popsicle box for the same amount of money. Similarly, if Pablo buys three single popsicles or both one 3-popsicle box and one single popsicle, he could have bought a 5-popsicle box for the same amount of money. If Pablo buys two 3-popsicle boxes, he could have bought a 5-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 5 -popsicle boxes, or a single 3 -popsicle box and the rest 5 -popsicle boxes. When Pablo has , he can obtain the maximum number of popsicles by buying two 5 -popsicle boxes and one 3 -popsicle box. This gives a total of popsicles.
The problems on this page are the property of the MAA's American Mathematics Competitions