The average age of 8 persons is increased by 2 years, when one of them, whose age is 24 years Is replaced by a new person. The age of the new person is :

Options

A. 42 years
B. 40 years
C. 38 years
D. 45 years

Answer

To find the age of the new person who replaces the 24-year-old, we can follow these steps:

### Step 1: Define the initial average age
Let the initial average age of the 8 persons be \( x \).

### Step 2: Calculate the initial total age
The total age of the 8 persons can be calculated as:
\[ \text{Total initial age} = 8 \times x = 8x \]

### Step 3: Define the new average age
According to the problem, the average age increases by 2 years when the 24-year-old is replaced. Therefore, the new average age is:
\[ \text{New average age} = x + 2 \]

### Step 4: Calculate the new total age
The total age of the 8 persons after the replacement is:
\[ \text{Total new age} = 8 \times (x + 2) = 8x + 16 \]

### Step 5: Set up the equation for the age increase
The total age increases from \( 8x \) to \( 8x + 16 \). The increase in total age is:
\[ \text{Increase in total age} = (8x + 16) - 8x = 16 \]

### Step 6: Relate the increase to the ages
The increase in total age of 16 years is due to the replacement of the 24-year-old with the new person. Therefore, we can set up the equation:
\[ \text{Age of new person} - 24 = 16 \]

### Step 7: Solve for the age of the new person
To find the age of the new person, we rearrange the equation:
\[ \text{Age of new person} = 16 + 24 = 40 \]

Thus, the age of the new person is **40 years**.

### Summary of Steps:
1. Define the initial average age as \( x \).
2. Calculate the initial total age as \( 8x \).
3. Define the new average age as \( x + 2 \).
4. Calculate the new total age as \( 8x + 16 \).
5. Find the increase in total age, which is 16.
6. Relate the increase to the ages of the persons.
7. Solve for the age of the new person.