The mean age of a combined group of men and women is 25 yrs . If mean age of men is 26 and that of women is 21 , then percentage of men and women in the group , is ,

Options

A. 60 , 40
B. 80 , 20
C. 20 , 80
D. 30 , 70

Explanation

To solve the problem, we will use the concept of weighted averages. Let's denote the number of men as \( m \) and the number of women as \( w \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Mean age of the combined group (men and women) = 25 years
- Mean age of men = 26 years
- Mean age of women = 21 years

2. **Set Up the Equation for the Mean**:
The mean age of the combined group can be expressed as:
\[ \text{Mean} = \frac{(m \times 26) + (w \times 21)}{m + w} = 25 \]

3. **Multiply Both Sides by \( m + w \)**:
This eliminates the denominator:
\[ (m \times 26) + (w \times 21) = 25(m + w) \]

4. **Expand the Right Side**:
\[ 26m + 21w = 25m + 25w \]

5. **Rearrange the Equation**:
Move all terms involving \( m \) and \( w \) to one side:
\[ 26m - 25m = 25w - 21w \]
Simplifying gives:
\[ m = 4w \]

6. **Express the Total Number of People**:
The total number of people in the group is:
\[ m + w = 4w + w = 5w \]

7. **Calculate the Percentages**:
- Percentage of men:
\[ \text{Percentage of men} = \left(\frac{m}{m + w}\right) \times 100 = \left(\frac{4w}{5w}\right) \times 100 = 80\% \]
- Percentage of women:
\[ \text{Percentage of women} = \left(\frac{w}{m + w}\right) \times 100 = \left(\frac{w}{5w}\right) \times 100 = 20\% \]

### Final Answer:
- Percentage of men = 80%
- Percentage of women = 20%