Question

In a numerical table with 10 rows and 10 columns, each entry is either a 9 or a 10. If the number of 9s 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? [Official GMAT-2018]

Hint:

To find the average of a set of numbers, calculate the total sum and divide by the total number of entries.

Answer 1

Step 1: Define the number of 9s in each row.
The number of 9s in the nth row is \( n - 1 \). Therefore, the number of 9s in each row is:
- 1st row: 0 9s, 10 10s
- 2nd row: 1 9, 9 10s
- 3rd row: 2 9s, 8 10s
- 4th row: 3 9s, 7 10s
- 5th row: 4 9s, 6 10s
- 6th row: 5 9s, 5 10s
- 7th row: 6 9s, 4 10s
- 8th row: 7 9s, 3 10s
- 9th row: 8 9s, 2 10s
- 10th row: 9 9s, 1 10
Step 2: Calculate the total number of 9s and 10s in the table.
- The total number of 9s is the sum of \( n - 1 \) from \( n = 1 \) to \( n = 10 \): \[ \text{Total number of 9s} = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \] - The total number of 10s is the remaining entries in each row (10 - \( n - 1 \)): \[ \text{Total number of 10s} = 10 \times 10 - 45 = 55 \] Step 3: Calculate the total sum of the numbers in the table.
- The sum of all the 9s is: \[ \text{Sum of 9s} = 9 \times 45 = 405 \] - The sum of all the 10s is: \[ \text{Sum of 10s} = 10 \times 55 = 550 \] Thus, the total sum of all numbers in the table is: \[ \text{Total sum} = 405 + 550 = 955 \] Step 4: Calculate the total number of entries in the table.
The table has 10 rows and 10 columns, so the total number of entries is: \[ \text{Total number of entries} = 10 \times 10 = 100 \] Step 5: Calculate the average.
The average is the total sum divided by the total number of entries: \[ \text{Average} = \frac{955}{100} = 9.55 \] Step 6: Conclusion.
The average of all the numbers in the table is 9.55.