In a numerical table with 10 rows and 10 columns, each entry is either a 9 or a 10 . If the...

Question

In a numerical table with 10 rows and 10 columns, each entry is either a 9 or a 10 . If the number of 9 s in the nth row is n-1 for each n from 1 to 10 , what is the average (arithmetic mean) of all the numbers in the table?

GMAT Panda · Accepted Answer

9.55 We first note that each entry is either "9" or "10", and that in the "nth row", the number of "9"s is "n - 1". Number of 9s in row n = n - 1 Number of 10s in row n = 11 - n We then calculate the sum of the entries in the "nth row". 9(n - 1) + 10(11 - n) = 101 - n We observe that the row sums from "n = 1" to "n = 10" 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 = 101 - 1 = 100 Last row sum = 101 - 10 = 91 Average row sum = 100 + 91/2 = 95.5 Total sum of all rows = 95.5 10 = 955 We then find the overall average by dividing the total sum by the total number of entries. Average per entry = 955/100 = 9.55

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$We first note that each entry is either "9" or "10", and that in the "nth row", the number of "9" s is "n - 1" .$	$Number of 9s in row n = n - 1 Number of 10s in row n = 11 - n$
$We then calculate the sum of the entries in the "nth row" .$	$9 (n - 1) + 10 (11 - n) = 101 - n$
$We observe that the row sums from "n = 1" to "n = 10" form an evenly spaced list, so by Average and Median Property of Evenly Spaced Sets the average of these sums is the mean of the first and last values.$	$First row sum = 101 - 1 = 100 Last row sum = 101 - 10 = 91 Average row sum = \frac{100 + 91}{2} = 95.5 Total sum of all rows = 95.5 \times 10 = 955$
$We then find the overall average by dividing the total sum by the total number of entries.$	$Average per entry = \frac{955}{100} = 9.55$

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