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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Solution:
The sum of the first two numbers is and the sum of the last two is , so the sum of all four is . The sum of the middle two is , so the sum of the first and last is , and their average is .
Let the four numbers be , and . It is given that
which implies that , , and . Adding the three equations yields , and substituting produces . Therefore the average of the first and last numbers is
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions