Question

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

• A. 30
• B. 40
• C. 50
• D. 60

Answer 1

The correct option is B 40

$Given,\overline{x}=$ Mean of Men and women $=30,$

${\overline{x}}_1=$ Mean of men $=32$

${\overline{x}}_2=$ Mean of women $=27$

$Let{n}_1+n_2=100$ where ${}^{\prime }{n}_1^{\prime }$ denotes number of men, ${}^{\prime }{n}_2^{\prime }$ denotes number of women for this, $n_2=100-n_1$

The formula for combined mean is

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2}{n_1+n_2}$

$\Rightarrow 30=\frac{32n_1+(100-n_1)27}{100}$

$\Rightarrow 30=\frac{32n_1+2700-27n_1}{100}$

$\Rightarrow 3000-2700=32n_1-27n_1$

$\Rightarrow 300=5n_1$

$\Rightarrow n_1=60$

$So,n_2=40$

Percentage of women $=\frac{\text{Number of women}}{\text{Total person}}\times 100\%$

$=\frac{40}{100}\times 100\%=40\%$

Hence, the percentage of women in the group is $40\%$