400 students took a mock exam in Delhi, 60% of the boys and 80% of the girls cleared the cut off in the examination. If the total percentage of students qualifying is 65%, how many girls appeared in the examination?

Options

A. 100
B. 120
C. 150
D. 300

Answer

To solve the problem step-by-step, we can follow these steps:

### Step 1: Understand the total number of students
We know that the total number of students who took the exam is 400.

### Step 2: Define variables for boys and girls
Let \( x \) be the number of girls who appeared for the examination. Therefore, the number of boys will be:
\[ 400 - x \]

### Step 3: Calculate the number of students who cleared the exam
According to the problem:
- 60% of the boys cleared the exam.
- 80% of the girls cleared the exam.

The number of boys who cleared the exam is:
\[ \text{Boys cleared} = 60\% \text{ of } (400 - x) = 0.6(400 - x) \]

The number of girls who cleared the exam is:
\[ \text{Girls cleared} = 80\% \text{ of } x = 0.8x \]

### Step 4: Set up the equation for total students who cleared the exam
The total number of students who cleared the exam is given by:
\[ \text{Total cleared} = \text{Boys cleared} + \text{Girls cleared} = 0.6(400 - x) + 0.8x \]

### Step 5: Use the overall percentage of students who cleared the exam
We know that 65% of the total 400 students cleared the exam:
\[ \text{Total cleared} = 65\% \text{ of } 400 = 0.65 \times 400 = 260 \]

### Step 6: Set up the equation
Now we can set up the equation:
\[ 0.6(400 - x) + 0.8x = 260 \]

### Step 7: Simplify the equation
Expanding the left side:
\[ 240 - 0.6x + 0.8x = 260 \]
Combining like terms:
\[ 240 + 0.2x = 260 \]

### Step 8: Solve for \( x \)
Subtract 240 from both sides:
\[ 0.2x = 20 \]
Now divide by 0.2:
\[ x = \frac{20}{0.2} = 100 \]

### Conclusion
The number of girls who appeared in the examination is:
\[ \boxed{100} \]