Average Marks Calculation: Finding the 7th Paper Marks

This problem involves calculating the marks obtained in a specific paper (the 7th paper) given the average marks of a larger set and two overlapping subsets.

Understanding the Given Information

We are provided with the following details:

• Total number of papers: 13
• Average mark of all 13 papers: 80
• Number of papers in the first group: 7
• Average mark of the first 7 papers: 84
• Number of papers in the last group: 7
• Average mark of the last 7 papers: 70

Step-by-Step Solution

1. Calculate Total Marks for All Papers

The total marks obtained in all 13 papers can be calculated using the formula:

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

For all 13 papers:

$$ \text{Total Marks}_{13} = 13 \times 80 = 1040 $$

2. Calculate Total Marks for the First 7 Papers

Similarly, the total marks for the first 7 papers are:

$$ \text{Total Marks}_{\text{first 7}} = 7 \times 84 $$

$$ \text{Total Marks}_{\text{first 7}} = 588 $$

3. Calculate Total Marks for the Last 7 Papers

The total marks for the last 7 papers are:

$$ \text{Total Marks}_{\text{last 7}} = 7 \times 70 $$

$$ \text{Total Marks}_{\text{last 7}} = 490 $$

4. Relate the Totals and Find the 7th Paper's Marks

The key insight here is that the 7th paper is included in *both* the 'first 7 papers' group and the 'last 7 papers' group. This means its marks are counted twice when we sum the totals of these two groups.

Let $M_7$ represent the marks obtained in the 7th paper.

The sum of marks for the first 7 papers includes papers 1 through 7.

The sum of marks for the last 7 papers includes papers 7 through 13.

When we add the total marks of the first 7 and the last 7 papers, we get:

$$ (\text{Marks of papers 1-6} + M_7) + (M_7 + \text{Marks of papers 8-13}) $$

This simplifies to:

$$ \text{Marks of papers 1-6} + \text{Marks of papers 8-13} + 2 \times M_7 $$

We know the total marks for all 13 papers is:

$$ \text{Total Marks}_{13} = \text{Marks of papers 1-6} + M_7 + \text{Marks of papers 8-13} $$

Therefore, the sum of the first 7 and last 7 papers can be expressed as:

$$ \text{Total Marks}_{\text{first 7}} + \text{Total Marks}_{\text{last 7}} = (\text{Total Marks}_{13} - M_7) + 2 \times M_7 $$

$$ \text{Total Marks}_{\text{first 7}} + \text{Total Marks}_{\text{last 7}} = \text{Total Marks}_{13} + M_7 $$

Now, we can rearrange this formula to solve for $M_7$:

$$ M_7 = (\text{Total Marks}_{\text{first 7}} + \text{Total Marks}_{\text{last 7}}) - \text{Total Marks}_{13} $$

5. Substitute Values and Calculate

Plugging in the calculated total marks:

$$ M_7 = (588 + 490) - 1040 $$

$$ M_7 = 1078 - 1040 $$

$$ M_7 = 38 $$

Summary of Marks

The marks obtained in the 7th paper are 38.