The average age of 8 men is increased by 2 years when two of them whose age are 21 and 23 years replaced by two new men. The average age of the two new men is

Options

A. 22 years
B. 24 years
C. 28 years
D. 30 years

Answer

To solve the problem step by step, we need to determine the average age of the two new men after replacing the two older men aged 21 and 23 years.

### Step 1: Calculate the initial total age of the 8 men.
Let the average age of the 8 men be \( x \). Therefore, the total age of the 8 men can be calculated as:
\[ \text{Total age of 8 men} = 8x \]

### Step 2: Calculate the total age after the replacement.
When two men aged 21 and 23 are replaced, their combined age is:
\[ 21 + 23 = 44 \text{ years} \]
The new average age of the 8 men is \( x + 2 \). Thus, the new total age of the 8 men becomes:
\[ \text{New total age of 8 men} = 8(x + 2) = 8x + 16 \]

### Step 3: Set up the equation for the total age before and after the replacement.
The new total age can also be expressed as the old total age minus the ages of the two men being replaced plus the ages of the two new men. Let the ages of the two new men be \( y_1 \) and \( y_2 \). Therefore, we can write:
\[ 8x + 16 = 8x - 44 + y_1 + y_2 \]

### Step 4: Simplify the equation.
By simplifying the equation, we get:
\[ 8x + 16 = 8x - 44 + y_1 + y_2 \]
Subtract \( 8x \) from both sides:
\[ 16 = -44 + y_1 + y_2 \]
Now, add 44 to both sides:
\[ 16 + 44 = y_1 + y_2 \]
\[ y_1 + y_2 = 60 \]

### Step 5: Calculate the average age of the two new men.
The average age of the two new men can be calculated as:
\[ \text{Average age of the two new men} = \frac{y_1 + y_2}{2} = \frac{60}{2} = 30 \]

### Final Answer:
The average age of the two new men is **30 years**.

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