Problem:
Andy and Bethany have a rectangular array of numbers with rows and columns. Andy adds the numbers in each row. The average of his sums is . Bethany adds the numbers in each column. The average of her sums is . What is the value of
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Solution:
There are rows, so the sum of the row sums is . This number is also the sum of all of the numbers in the array because each number in the array is added to obtain one of the row sums. Similarly, there are columns, so the sum of the column sums is , and this, too, is the sum of all of the numbers in the array. So , and .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions