A bus starts from a village P having some number of boys and girls. The number of total girls is 1/4th of the number of total boys. At village Q, 20 boys left the bus and 10 girls are joined. After this the number of boys will equal the number of girls, then find the total initial number of boys and girls.

Options

A. 40
B. 20
C. 25
D. 50

Answer

To solve the problem, we need to set up equations based on the information provided in the question.

### Step-by-Step Solution:

1. **Define Variables**:
Let the number of boys be \( x \) and the number of girls be \( y \).

2. **Set Up the Relationship**:
According to the problem, the number of girls is \( \frac{1}{4} \) of the number of boys. This can be expressed as:
\[ y = \frac{1}{4}x \]

3. **Changes at Village Q**:
At village Q, 20 boys leave the bus and 10 girls join. After these changes, the number of boys and girls becomes equal. The new number of boys will be \( x - 20 \) and the new number of girls will be \( y + 10 \).

4. **Set Up the Equality**:
After the changes, we have:
\[ x - 20 = y + 10 \]

5. **Substitute the Value of \( y \)**:
Substitute \( y \) from step 2 into the equation from step 4:
\[ x - 20 = \frac{1}{4}x + 10 \]

6. **Solve for \( x \)**:
To eliminate the fraction, multiply the entire equation by 4:
\[ 4(x - 20) = x + 40 \]
Expanding gives:
\[ 4x - 80 = x + 40 \]
Rearranging the equation:
\[ 4x - x = 40 + 80 \]
\[ 3x = 120 \]
\[ x = 40 \]

7. **Find \( y \)**:
Now substitute \( x \) back into the equation for \( y \):
\[ y = \frac{1}{4} \times 40 = 10 \]

8. **Total Initial Number of Boys and Girls**:
The total initial number of boys and girls is:
\[ x + y = 40 + 10 = 50 \]

### Final Answer:
The total initial number of boys and girls is **50**.