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

Options

A. 46, 60
B. 80, 20
C. 20, 80
D. 60, 40

Answer

To solve the problem, we need to find the percentage of men and women in a combined group given their mean ages. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( m \) = number of men
- \( w \) = number of women

### Step 2: Set Up the Equations
We know the following:
- The mean age of the combined group (men and women) is 25 years.
- The mean age of men is 26 years.
- The mean age of women is 21 years.

The total age of the men can be expressed as \( 26m \) (since the mean age of men is 26), and the total age of the women can be expressed as \( 21w \) (since the mean age of women is 21).

The mean age of the combined group can be represented as:
\[ \frac{26m + 21w}{m + w} = 25 \]

### Step 3: Cross Multiply to Eliminate the Fraction
Cross-multiplying gives us:
\[ 26m + 21w = 25(m + w) \]

### Step 4: Expand and Rearrange the Equation
Expanding the right side:
\[ 26m + 21w = 25m + 25w \]

Now, rearranging the equation:
\[ 26m - 25m = 25w - 21w \]
\[ m = 4w \]

### Step 5: Express in Terms of Total Population
Let’s assume the total number of people in the group is \( N = m + w \). From the equation \( m = 4w \), we can substitute \( m \):
\[ N = 4w + w = 5w \]

### Step 6: Find the Values of \( m \) and \( w \)
From \( N = 5w \):
\[ w = \frac{N}{5} \quad \text{and} \quad m = 4w = 4 \left(\frac{N}{5}\right) = \frac{4N}{5} \]

### Step 7: Calculate Percentages
Now, we can find the percentages of men and women:
- Percentage of men:
\[ \text{Percentage of men} = \left(\frac{m}{N}\right) \times 100 = \left(\frac{4N/5}{N}\right) \times 100 = 80\% \]

- Percentage of women:
\[ \text{Percentage of women} = \left(\frac{w}{N}\right) \times 100 = \left(\frac{N/5}{N}\right) \times 100 = 20\% \]

### Final Answer
The percentage of men in the group is **80%** and the percentage of women is **20%**.

---