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?

1. 9.45
2. 9.50
3. 9.55
4. 9.65
5. 9.70

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

The problem statement states that:

Given:

• In a numerical table with 10 rows and 10 columns, each entry is either a 9 or a 10.
• The number of 9s in the nth row is n – 1 for each n from 1 to 10.

Find out:

• The average (arithmetic mean) of all the numbers in the table

We can see that there are ten rows and columns, and all are filled with entries.
Thus there are 10 X 10 entries= 100 entries.

The hypothesis is that the number of 9s in the nth row of the table is equal to n - 1 for each n from 1 to 10. This means that:
In the 10th row, the number of 9s is 10 - 1 = 9.
In the 9th row, the number of 9s is 9 - 1 = 8
In the 8th row, the number of 9s is 8 - 1 = 7
Similarly, for the 7th row, the number of 9s is 7 - 1 = 6
If we go down to the last row, that is the first row then the number of nines is 1-1=0.

Hence the total number of 9s is 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 9*(9 + 1)/2 = 45.

The remaining entries will equal the number of tens, which can be calculated by deducting 45 from 100.
Accordingly, the number of tens is equal to 100-45, or 55.
The average of all numbers= (45 x 9 + 55 x 10)/100 = 9.55

This image is given below for a better understanding:

Approach Solution 2:

The problem statement suggests that:

Given:

• In a numerical table with 10 rows and 10 columns, each entry is either a 9 or a 10.
• The number of 9s in the nth row is n – 1 for each n from 1 to 10.

Find out:

• The average (arithmetic mean) of all the numbers in the table.

A numbered table with ten rows and ten columns is given in the question.
There are entries in every one of the rows and columns.
There are two possible entries in the table: 9 and 10.

The hypothesis in the question further indicates that for each n from 1 to 10 in the table.
The number of 9s in the nth row is equal to n - 1.
Assume all numbers were 10, then the sum would have been 10*10(rows)*10(columns) = 1000.

Now, this sum is reduced by including some 9s in the rows in a specific pattern. Thus, subtract the extra counted 1 (10 – 9 = 1) from the above sum.
There are 0+1+2+3+4+5+6+7+8+9 = 45 9s.
Consequently, deduct 45 from 1000.
The result is 955, which is the sum of all numbers on the grid.
Hence, average = 955/100 = 9.55

Approach Solution 3:

The problem statement implies that:

Given:

• In a numerical table with 10 rows and 10 columns, each entry is either a 9 or a 10.
• The number of 9s in the nth row is n – 1 for each n from 1 to 10.

Find out:

• The average (arithmetic mean) of all the numbers in the table.

Since the table has 10 rows and 10 columns there are 100 spots in the table.
Therefore, we see there are:
For row 1: n = 1 and there are (n - 1) = (1 - 1) = 0.
Hence, there are 0 nines in row 1

For row 2: n = 2 and there are (n - 1) = (2 - 1) = 1.
Hence, there is 1 nine in row 2.

For row 3: n = 3 and there are (n - 1) = (3 - 1) = 2.
Hence, there are 2 nines in row 3, and so on.

We can see that a pattern emerges.
Therefore, the number of nines is = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Since there are 100 numbers in the entire table, we have (100 - 45) = 55 tens.
Therefore, the average of the 45 nines and 55 tens is
= [9(45) + 10(55)]/100
= (405 + 550)/100
= 9.55
Hence, the average (arithmetic mean) of all the numbers in the table = 9.55

“In a numerical table with 10 rows and 10 columns”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This book has been taken from the book “GMAT Official Guide Quantitative Review 2022”. To solve the GMAT Problem Solving questions, the candidates must have basic concepts of mathematics. The candidates can go through GMAT Quant practice papers to practice varieties of questions that will enable them to improve their mathematical skills.

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