Average age of a group of 30 boys is 16 years. A boy of age 19 leaves the group and a new boy joins the group. If the new average age of the group is 15.8 years, find the age of the new boy.

Options

A. A. 15 years
B. B. 14 years
C. C. 13 years
D. D. 12 years

Answer

To find the age of the new boy who joins the group after a boy of age 19 leaves, we can follow these steps:

### Step 1: Calculate the total age of the original group
The average age of the group of 30 boys is 16 years. Therefore, the total age of the group can be calculated as:
\[ \text{Total age} = \text{Average age} \times \text{Number of boys} = 16 \times 30 = 480 \text{ years} \]

### Step 2: Calculate the total age after the boy of age 19 leaves
When the boy aged 19 leaves the group, the total age of the group becomes:
\[ \text{New total age} = \text{Total age} - \text{Age of the boy leaving} = 480 - 19 = 461 \text{ years} \]

### Step 3: Calculate the new average age after the boy leaves
After the boy leaves, there are now 29 boys in the group. We know that the new average age is 15.8 years. Therefore, the total age of the group with the new boy can be expressed as:
\[ \text{New total age} = \text{New average age} \times \text{Number of boys} = 15.8 \times 30 = 474 \text{ years} \]

### Step 4: Calculate the age of the new boy
Let \( x \) be the age of the new boy. The total age after the new boy joins can be expressed as:
\[ \text{Total age with new boy} = \text{New total age after boy leaves} + x = 461 + x \]
Setting this equal to the new total age calculated:
\[ 461 + x = 474 \]
Solving for \( x \):
\[ x = 474 - 461 = 13 \]

### Conclusion
The age of the new boy is **13 years**.