What number is placed in the shaded circle if each of the numbers is placed in a different circle, the numbers 13 and 17 are placed as shown, and Jen calculates the average of the numbers in the first three circles, the average of the numbers in the middle three circles, and the average of the numbers in the last three circles, and these three averages are equal?
Solution
The averages of groups of three numbers are equal if the sums of the numbers in each group are equal, because in each case the average equals the sum of the three numbers divided by 3. Therefore, the averages of three groups of three numbers are equal if the sum of each of the three groups are equal.
The original nine numbers have a sum of
and so if these are divided into three groups of equal sum, the sum of each group is . Consider the middle three numbers. Since two of the numbers are 13 and 17, then the third number must be . We note that the remaining six numbers can be split into the groups and , each of which also has a sum of 37.
Therefore, the number that is placed in the shaded circle is 7.
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