600 students took a mock exam in Delhi. `55%` of the boys and `75%` of the girls cleared the cut off in the examination. If the total percentage of students qualifying is `63%`. Find the number of boys and girls appeared in the examination.

Answer

To solve the problem, we will use the concept of alligation to find the number of boys and girls who appeared in the examination.

### Step-by-Step Solution

1. **Identify the total number of students**:
The total number of students who took the mock exam is given as 600.

2. **Define the percentages**:
- Percentage of boys who cleared the cutoff: 55%
- Percentage of girls who cleared the cutoff: 75%
- Overall percentage of students who cleared the cutoff: 63%

3. **Set up the alligation**:
We will use the alligation method to find the ratio of boys to girls.
- Let the percentage of boys who cleared the cutoff be \( B = 55\% \).
- Let the percentage of girls who cleared the cutoff be \( G = 75\% \).
- Let the overall percentage of students who cleared the cutoff be \( P = 63\% \).

The alligation formula gives us:
\[ \text{Ratio of boys to girls} = \frac{G - P}{P - B} \]

4. **Calculate the differences**:
- \( G - P = 75 - 63 = 12 \)
- \( P - B = 63 - 55 = 8 \)

5. **Form the ratio**:
Thus, the ratio of boys to girls is:
\[ \text{Ratio of boys to girls} = \frac{12}{8} = \frac{3}{2} \]

6. **Express boys and girls in terms of a variable**:
Let the number of boys be \( 3k \) and the number of girls be \( 2k \).

7. **Set up the equation**:
Since the total number of students is 600, we can write:
\[ 3k + 2k = 600 \]
\[ 5k = 600 \]

8. **Solve for \( k \)**:
Dividing both sides by 5:
\[ k = \frac{600}{5} = 120 \]

9. **Calculate the number of boys and girls**:
- Number of boys = \( 3k = 3 \times 120 = 360 \)
- Number of girls = \( 2k = 2 \times 120 = 240 \)

### Final Answer
- Number of boys who appeared in the examination: **360**
- Number of girls who appeared in the examination: **240**