Problem:
, and are three piles of rocks. The mean weight of the rocks in is 40 pounds, the mean weight of the rocks in is 50 pounds, the mean weight of the rocks in the combined piles and is 43 pounds, and the mean weight of the rocks in the combined piles and is 44 pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles and ?
Answer Choices:
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B.
C.
D.
E.
Solution:
Let , and be the number of rocks in piles , and , respectively. Then
Because 7 and 3 are relatively prime, there is a positive integer such that and . Let equal the mean weight in pounds of the rocks in and equal the mean weight in pounds of the rocks in and . Then
and
Clearing the denominator and rearranging yields . Because the mean weight of the rocks in the combined piles and is 44 pounds, and the mean weight of the rocks in is greater than the mean weight of the rocks in , it follows that the mean weight of the rocks in and must be greater than 44 pounds. Thus and therefore must be greater than zero. This implies that . If and , then . Thus the greatest possible integer value for the weight in pounds of the combined piles and is .
The problems on this page are the property of the MAA's American Mathematics Competitions