The mean age of a combined group of men and women is 30 years. If the means of the age of men and women are respectively 32 and 27, then the percentage of women in the group is

Options

A. 30
B. 40
C. 50
D. 60

Answer

To solve the problem step by step, we can follow these calculations:

**Step 1: Define the variables.**
Let the percentage of men in the group be \( X \% \). Therefore, the percentage of women in the group will be \( 100 - X \% \).

**Step 2: Write the equation for the mean age.**
The mean age of the combined group can be expressed as:
\[ \text{Mean Age} = \frac{(\text{Mean Age of Men} \times \text{Percentage of Men}) + (\text{Mean Age of Women} \times \text{Percentage of Women})}{\text{Total Percentage}} \]
Substituting the known values:
\[ 30 = \frac{(32 \times X) + (27 \times (100 - X))}{100} \]

**Step 3: Simplify the equation.**
Multiply both sides by 100 to eliminate the denominator:
\[ 3000 = 32X + 27(100 - X) \]
Distributing the 27:
\[ 3000 = 32X + 2700 - 27X \]

**Step 4: Combine like terms.**
Combine the terms involving \( X \):
\[ 3000 = (32X - 27X) + 2700 \]
\[ 3000 = 5X + 2700 \]

**Step 5: Isolate \( X \).**
Subtract 2700 from both sides:
\[ 3000 - 2700 = 5X \]
\[ 300 = 5X \]
Now, divide both sides by 5:
\[ X = \frac{300}{5} = 60 \]

**Step 6: Calculate the percentage of women.**
Since \( X \) is the percentage of men, the percentage of women is:
\[ 100 - X = 100 - 60 = 40 \]

**Final Answer:**
The percentage of women in the group is \( 40\% \).

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