S95-11

Official Solution:

At a delivery store, seven packages have an average (arithmetic mean) weight of 225 pounds and a median weight of 270 pounds. What is the maximum possible weight, in pounds, of the lightest package?

A. 25
B. 165
C. 195
D. 225
E. 270


Since the mean is fixed at 225 pounds, we can maximize the weight of the lightest package(s) by choosing the smallest possible weight for the heaviest packages. Since the median is 270 pounds and there are seven packages, we know the weight of the fourth largest package must be 270 pounds. So, we can further deduce that the smallest possible weight for each of the additional three heaviest packages is 270 pounds.

If we subtract the weight of the heaviest four packages from the total weight of all seven packages (remembering that \(\text{total } = \text{ mean } \times \text{ number of items}\)), we can calculate the total weight for the lightest three packages: \(225 \text{ pounds } \times 7 - 270 \text{ pounds } \times 4 = 1,575 \text{ pounds } - 1,080 \text{ pounds } = 495 \text{ pounds}\).

We want the lightest of the three smaller packages to have the highest weight possible. To accomplish this, we need to give all three of these smaller packages the same weight. So, to find the maximum possible weight of the lightest package, we divide 495 by 3 to get 165.


Answer: B