The average mark of 13 papers is 80. The average marks of the first 7 papers are 84 and that of the last 7 papers is 70. Find the marks obtained in the 7th paper.

Correct Answer: 38

Step-by-Step Explanation:

Step 1: Understand the formula for the sum of marks

The average of a set of numbers is given by the total sum divided by the total count of numbers:

$\text{Average}=\frac{\text{Total Sum of Marks}}{\text{Number of Papers}}$

From this, we can find the total sum by multiplying the average by the number of papers:

$\text{Total Sum of Marks}=\text{Average}\times \text{Number of Papers}$

Step 2: Calculate the total marks for all 13 papers

Given that the average mark of 13 papers is 80:

${\text{Sum}}_{\text{total}}=13\times 80=1040$

Step 3: Calculate the total marks for the first 7 papers

Given that the average mark of the first 7 papers is 84:

${\text{Sum}}_{\text{first 7}}=7\times 84=588$

Step 4: Calculate the total marks for the last 7 papers

Given that the average mark of the last 7 papers is 70:

${\text{Sum}}_{\text{last 7}}=7\times 70=490$

Step 5: Find the marks obtained in the 7th paper

When we add the total marks of the first 7 papers and the last 7 papers, the 7th paper is included in both groups (counted twice). Therefore, the sum of the first 7 papers plus the last 7 papers is equal to the total marks of all 13 papers plus the marks of the 7th paper.

${\text{Sum}}_{\text{first 7}}+{\text{Sum}}_{\text{last 7}}={\text{Sum}}_{\text{total}}+\text{Marks in 7th paper}$

Rearranging the equation to solve for the 7th paper's marks:

$\text{Marks in 7th paper}=({\text{Sum}}_{\text{first 7}}+{\text{Sum}}_{\text{last 7}})-{\text{Sum}}_{\text{total}}$

Substitute the calculated values into the equation:

$\text{Marks in 7th paper}=(588+490)-1040$

$\text{Marks in 7th paper}=1078-1040=38$

Thus, the marks obtained in the 7th paper are 38.