GMAT Practice Question: 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?
- 9.45
- 9.50
- 9.55
- 9.65
- 9.70
Topics: word problems, mean
Solution
Step 1
We first note that each entry is either \textbf{"9"} or \textbf{"10"}, and that in the \textbf{"nth row"}, the number of \textbf{"9"}s is \textbf{"n - 1"}.
\text{Number of 9s in row }n = n - 1 \text{Number of 10s in row }n = 11 - n
Step 2
We then calculate the sum of the entries in the \textbf{"nth row"}.
9(n - 1) + 10(11 - n) = 101 - n
Step 3
We observe that the row sums from \textbf{"n = 1"} to \textbf{"n = 10"} form an evenly spaced list, so by \textbf{Average and Median Property of Evenly Spaced Sets} the average of these sums is the mean of the first and last values.
\text{First row sum} = 101 - 1 = 100 \text{Last row sum} = 101 - 10 = 91 \text{Average row sum} = \frac{100 + 91}{2} = 95.5 \text{Total sum of all rows} = 95.5 \times 10 = 955
Step 4
We then find the overall average by dividing the total sum by the total number of entries.
\text{Average per entry} = \frac{955}{100} = 9.55
Answer
9.55
