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In a 9 \times 9 table, each entry is either a 7 or an 8 . If the number of 7 s in the n th row is 9 - n for each n from 1 to 9, what is the average (arithmetic mean) of all the numbers in the table?

Explanation	Calculations	Help
$We first note that each entry is either "7" or "8", and that in the "nth row", the number of "7" s is "9 - n" .$	$Number of 7s in row n = 9 - n Number of 8s in row n = 9 - (9 - n) = n$
$We then calculate the sum of the entries in the "nth row" .$	$7 \times (9 - n) + 8 n = 63 + n$
$We observe that the row sums from "n = 1" to "n = 9" form an evenly spaced list, so by "Average and Median Property of Evenly Spaced Sets" the average of these sums is the mean of the first and last values.$	$First row sum = 63 + 1 = 64 Last row sum = 63 + 9 = 72 Average row sum = \frac{64 + 72}{2} = 68 Total sum of all rows = 68 \times 9 = 612$	Theory & Tactics Method Card EVEN1-C Click to view full details
$We then find the overall average by dividing the total sum by the total number of entries.$	$\frac{612}{81} = \frac{68}{9} \frac{68}{9} = 7 + \frac{5}{9} 7 + \frac{5}{9} \approx 7.56$	Theory & Tactics Method Card CALC2-E Click to view full details

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C

In a 9 \times 9 table, each entry is either a 7 or an 8 . If the number of 7 s in the n th row is 9 - n for each n from 1 to 9, what is the average (arithmetic mean) of all the numbers in the table?

Explanation	Calculations	Help
$We first note that each entry is either "7" or "8", and that in the "nth row", the number of "7" s is "9 - n" .$	$Number of 7s in row n = 9 - n Number of 8s in row n = 9 - (9 - n) = n$
$We then calculate the sum of the entries in the "nth row" .$	$7 \times (9 - n) + 8 n = 63 + n$
$We observe that the row sums from "n = 1" to "n = 9" form an evenly spaced list, so by "Average and Median Property of Evenly Spaced Sets" the average of these sums is the mean of the first and last values.$	$First row sum = 63 + 1 = 64 Last row sum = 63 + 9 = 72 Average row sum = \frac{64 + 72}{2} = 68 Total sum of all rows = 68 \times 9 = 612$	Theory & Tactics Method Card EVEN1-C Click to view full details
$We then find the overall average by dividing the total sum by the total number of entries.$	$\frac{612}{81} = \frac{68}{9} \frac{68}{9} = 7 + \frac{5}{9} 7 + \frac{5}{9} \approx 7.56$	Theory & Tactics Method Card CALC2-E Click to view full details

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C