The average age of a group of persons going for a picnic is 16.75 years. 20 new persons with an average age of 13.25 years join the group on the spot due to which the average of the group becomes 15 years. Find the number of persons initially going for the picnic.

Options

A. 24
B. 20
C. 15
D. 18

Answer

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

1. **Let the initial number of persons be \( X \)**:
We denote the initial number of persons going for the picnic as \( X \).

2. **Calculate the total age of the initial group**:
The average age of the initial group is given as 16.75 years. Therefore, the total age of the initial group can be calculated as:
\[ \text{Total age of initial group} = \text{Average age} \times \text{Number of persons} = 16.75 \times X \]

3. **Calculate the total age of the new persons joining**:
We know that 20 new persons join the group with an average age of 13.25 years. The total age of these new persons is:
\[ \text{Total age of new persons} = \text{Average age} \times \text{Number of new persons} = 13.25 \times 20 = 265 \]

4. **Calculate the new total age of the group**:
After the new persons join, the total age of the group becomes:
\[ \text{New total age} = \text{Total age of initial group} + \text{Total age of new persons} = 16.75X + 265 \]

5. **Calculate the new total number of persons in the group**:
The new total number of persons in the group is:
\[ \text{New total number of persons} = X + 20 \]

6. **Set up the equation for the new average age**:
The new average age of the group is given as 15 years. Therefore, we can set up the equation:
\[ \frac{16.75X + 265}{X + 20} = 15 \]

7. **Cross-multiply to eliminate the fraction**:
By cross-multiplying, we get:
\[ 16.75X + 265 = 15(X + 20) \]

8. **Expand the right side of the equation**:
Expanding the right side gives:
\[ 16.75X + 265 = 15X + 300 \]

9. **Rearrange the equation to isolate \( X \)**:
Subtract \( 15X \) from both sides:
\[ 16.75X - 15X + 265 = 300 \]
This simplifies to:
\[ 1.75X + 265 = 300 \]

10. **Subtract 265 from both sides**:
\[ 1.75X = 300 - 265 \]
\[ 1.75X = 35 \]

11. **Solve for \( X \)**:
Divide both sides by 1.75:
\[ X = \frac{35}{1.75} = 20 \]

Thus, the number of persons initially going for the picnic is \( \boxed{20} \).