Problem:
Four numbers are written in a row. The average of the first two is , the average of the middle two is , and the average of the last two is . What is the average of the first and last of the numbers?
Answer Choices:
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E.
Solution:
Let the numbers be in order. Since the average of and is , we know their sum is . Since the average of and is , we know their sum is . Since and , we know . Now, with the average of and being , we know their sum is . This means . Subtracting this result from the sum of all the terms yields . Since , our answer is .
Thus, the answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions