If a 12 years old boy is replaced by a new boy, then the average age of 18 boys increases by 1.5 years. What is the age of the new boy?

Options

A. 35 years
B. 37.5 years
C. 43 years
D. 39 years

Answer

To solve the problem step by step, we will follow the information given in the question and apply some basic algebra.

### Step 1: Understand the problem
We know that the average age of 18 boys increases by 1.5 years when a 12-year-old boy is replaced by a new boy. We need to find the age of the new boy.

### Step 2: Define variables
Let the initial average age of the 18 boys be \( x \). Therefore, the total age of the 18 boys can be expressed as:
\[ \text{Total age of 18 boys} = 18x \]

### Step 3: Calculate the new average
When the 12-year-old boy is replaced by the new boy, the new average age becomes \( x + 1.5 \). Thus, the total age of the 18 boys after the replacement can be expressed as:
\[ \text{New total age} = 18(x + 1.5) \]

### Step 4: Set up the equation
The new total age can also be expressed as the old total age minus the age of the boy who left (12 years) plus the age of the new boy (let's denote it as \( y \)):
\[ 18x - 12 + y \]

### Step 5: Equate the two expressions for total age
Now we can set the two expressions for the total age equal to each other:
\[ 18x - 12 + y = 18(x + 1.5) \]

### Step 6: Simplify the equation
Expanding the right side gives:
\[ 18x - 12 + y = 18x + 27 \]

### Step 7: Cancel out \( 18x \)
Subtract \( 18x \) from both sides:
\[ -12 + y = 27 \]

### Step 8: Solve for \( y \)
Now, add 12 to both sides to solve for \( y \):
\[ y = 27 + 12 \]
\[ y = 39 \]

### Conclusion
The age of the new boy is \( 39 \) years.