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In an 8 \times 8 table, each entry is either an 8 or a 12 . If the number of 8's in the n th row is n - 1 for each integer n from 1 to 8, what is the average (arithmetic mean) of all the numbers in the table?

Explanation	Calculations	Help
$We first note that each entry is either "8" or "12", and that in the "nth row", the number of "8" s is "n - 1" .$	$Number of 8s in row n = n - 1 Number of 12s in row n = 9 - n$
$We then calculate the sum of the entries in the "nth row" .$	$8 (n - 1) + 12 (9 - n) = 8 n - 8 + 108 - 12 n = - 4 n + 100$
$We observe that the row sums from "n = 1" to "n = 8" form an evenly spaced list, so by "Average and Median Property of Evenly Spaced Sets" the average of these sums is the mean of the first and last values.$	$First row sum = 100 - 4 \times 1 = 96 Last row sum = 100 - 4 \times 8 = 68 Average row sum = \frac{96 + 68}{2} = 82 Total sum of all rows = 82 \times 8 = 656$	Theory & Tactics Method Card EVEN1-C Click to view full details
$We then find the overall average by dividing the total sum by the total number of entries.$	$Average per entry = \frac{656}{64} = \frac{82}{8} = 10.25$

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C

In an 8 \times 8 table, each entry is either an 8 or a 12 . If the number of 8's in the n th row is n - 1 for each integer n from 1 to 8, what is the average (arithmetic mean) of all the numbers in the table?

Explanation	Calculations	Help
$We first note that each entry is either "8" or "12", and that in the "nth row", the number of "8" s is "n - 1" .$	$Number of 8s in row n = n - 1 Number of 12s in row n = 9 - n$
$We then calculate the sum of the entries in the "nth row" .$	$8 (n - 1) + 12 (9 - n) = 8 n - 8 + 108 - 12 n = - 4 n + 100$
$We observe that the row sums from "n = 1" to "n = 8" form an evenly spaced list, so by "Average and Median Property of Evenly Spaced Sets" the average of these sums is the mean of the first and last values.$	$First row sum = 100 - 4 \times 1 = 96 Last row sum = 100 - 4 \times 8 = 68 Average row sum = \frac{96 + 68}{2} = 82 Total sum of all rows = 82 \times 8 = 656$	Theory & Tactics Method Card EVEN1-C Click to view full details
$We then find the overall average by dividing the total sum by the total number of entries.$	$Average per entry = \frac{656}{64} = \frac{82}{8} = 10.25$

Scroll horizontally to view all columns

C