The average age of 11 persons in a committee is increased by 2 years when three men aged 32 years, 33 years and 34 years are substituted by three women. Find the average age of these three women

Answer

To solve the problem step by step, we need to find the average age of the three women who replaced the three men in the committee.

### Step 1: Calculate the total age of the committee before the replacement.
Let the average age of the 11 persons in the committee be \( x \). Therefore, the total age of the committee before the replacement is:
\[ \text{Total age before} = 11x \]

### Step 2: Calculate the total age of the three men.
The ages of the three men are 32 years, 33 years, and 34 years. Therefore, the total age of the three men is:
\[ \text{Total age of men} = 32 + 33 + 34 = 99 \text{ years} \]

### Step 3: Calculate the total age of the committee after the replacement.
After replacing the three men with three women, the average age of the committee increases by 2 years. Therefore, the new average age of the committee becomes \( x + 2 \). The total age of the committee after the replacement is:
\[ \text{Total age after} = 11(x + 2) = 11x + 22 \]

### Step 4: Set up the equation for the total ages.
Since the total age of the committee after the replacement can also be expressed as the total age before the replacement minus the total age of the men plus the total age of the women, we can write:
\[ 11x + 22 = 11x - 99 + \text{Total age of women} \]

### Step 5: Simplify the equation.
By simplifying the equation, we can isolate the total age of the women:
\[ 11x + 22 = 11x - 99 + \text{Total age of women} \]
\[ 22 + 99 = \text{Total age of women} \]
\[ \text{Total age of women} = 121 \text{ years} \]

### Step 6: Calculate the average age of the three women.
The average age of the three women can be calculated by dividing the total age of the women by the number of women:
\[ \text{Average age of women} = \frac{\text{Total age of women}}{3} = \frac{121}{3} \approx 40.33 \text{ years} \]

### Final Answer:
The average age of the three women is approximately \( 40 \frac{1}{3} \) years or \( 40 \) years and \( 4 \) months.

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