There were five sections in MAT paper.the average score of pooja in first 3 sections was 83 and the average in the last 3 sections was 97 and average of all the sections(i.e.,whole paper) was 92 then her score in the third section was:

Options

A. 85
B. 92
C. 88
D. none of these

Answer

To solve the problem step by step, we will denote the scores in the five sections as follows:

- Let \( S_1 \) be the score in the first section.
- Let \( S_2 \) be the score in the second section.
- Let \( S_3 \) be the score in the third section.
- Let \( S_4 \) be the score in the fourth section.
- Let \( S_5 \) be the score in the fifth section.

### Step 1: Calculate the total score for the first three sections
The average score of Pooja in the first three sections is 83. Therefore, we can express this as:
\[ \frac{S_1 + S_2 + S_3}{3} = 83 \]
Multiplying both sides by 3 gives:
\[ S_1 + S_2 + S_3 = 249 \quad \text{(Equation 1)} \]

### Step 2: Calculate the total score for the last three sections
The average score of Pooja in the last three sections is 97. Therefore, we can express this as:
\[ \frac{S_3 + S_4 + S_5}{3} = 97 \]
Multiplying both sides by 3 gives:
\[ S_3 + S_4 + S_5 = 291 \quad \text{(Equation 2)} \]

### Step 3: Calculate the total score for all five sections
The average score of Pooja in all five sections is 92. Therefore, we can express this as:
\[ \frac{S_1 + S_2 + S_3 + S_4 + S_5}{5} = 92 \]
Multiplying both sides by 5 gives:
\[ S_1 + S_2 + S_3 + S_4 + S_5 = 460 \quad \text{(Equation 3)} \]

### Step 4: Substitute and solve for \( S_3 \)
Now we have three equations:
1. \( S_1 + S_2 + S_3 = 249 \) (Equation 1)
2. \( S_3 + S_4 + S_5 = 291 \) (Equation 2)
3. \( S_1 + S_2 + S_3 + S_4 + S_5 = 460 \) (Equation 3)

From Equation 3, we can express \( S_4 + S_5 \) in terms of \( S_1 + S_2 + S_3 \):
\[ S_4 + S_5 = 460 - (S_1 + S_2 + S_3) \]
Substituting Equation 1 into this gives:
\[ S_4 + S_5 = 460 - 249 = 211 \quad \text{(Equation 4)} \]

### Step 5: Substitute Equation 4 into Equation 2
Now, we can substitute Equation 4 into Equation 2:
\[ S_3 + S_4 + S_5 = 291 \]
Substituting \( S_4 + S_5 = 211 \) gives:
\[ S_3 + 211 = 291 \]
Solving for \( S_3 \):
\[ S_3 = 291 - 211 = 80 \]

### Final Answer
Pooja's score in the third section is \( S_3 = 80 \).

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