The average age of 8 persons in a committee is increased by 2 years when two men aged 35 years and 45 years are substituted by two women. The average age of these two women is

Options

A. (a) 41 years
B. (b) 45 years
C. (c) 48 years
D. (d) 53 years

Answer

To solve the problem step by step, we need to follow the information given in the question about the average age of the committee members and the changes made.

### Step 1: Define the initial average age
Let the initial average age of the 8 persons in the committee be \( x \). Therefore, the total age of the 8 persons can be expressed as:
\[ \text{Total age of 8 persons} = 8x \]

**Hint:** Remember that the total age can be calculated by multiplying the average age by the number of persons.

### Step 2: Calculate the total age after substitution
When two men aged 35 years and 45 years are removed, their combined age is:
\[ 35 + 45 = 80 \text{ years} \]
After removing these two men, the total age of the remaining 6 persons is:
\[ 8x - 80 \]

**Hint:** Always sum the ages of the individuals being removed to find the new total.

### Step 3: New average age after substitution
The average age of the committee increases by 2 years, so the new average age becomes:
\[ x + 2 \]
Thus, the total age of the 8 persons after adding the two women can be expressed as:
\[ \text{Total age after substitution} = 8(x + 2) = 8x + 16 \]

**Hint:** When the average increases, you can express the new total age as the new average multiplied by the number of persons.

### Step 4: Set up the equation
Now, we can set up the equation based on the total ages before and after the substitution:
\[ 8x - 80 + Y = 8x + 16 \]
where \( Y \) is the total age of the two women.

**Hint:** Equate the total age before and after the substitution to find the unknown.

### Step 5: Solve for Y
Rearranging the equation gives:
\[ Y = (8x + 16) - (8x - 80) \]
\[ Y = 16 + 80 = 96 \]

**Hint:** Simplifying the equation helps isolate the variable you're trying to find.

### Step 6: Calculate the average age of the two women
The average age of the two women can be calculated as:
\[ \text{Average age of 2 women} = \frac{Y}{2} = \frac{96}{2} = 48 \]

**Hint:** To find the average, divide the total age by the number of individuals.

### Final Answer
The average age of the two women is **48 years**.