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 $32\text{ and }27$ , then find out the percentage of women in the group?
A. $30$
B. $40$
C. $50$
D. $60$

Answer 1

We will first assume the number of men and women and then we will find the sum of age of total men and then the sum of age of total women using the formula: $\text{Average}=\dfrac{\text{Sum of observation}}{\text{Total number of observation}}$ and then finally we will find the ratio of men and women using the same formula but on both the men and women and then we will find the percent of women by applying $n\%=\left( \dfrac{n}{m+n} \right)\times 100$ and $n$ is from the ratio $m:n$ .

Complete step-by-step answer :
Let the number of men in the group is $x$ and the number of women in the group be $y$ ,
Now we know that the formula for average is as following:
$\text{Average}=\dfrac{\text{Sum of observation}}{\text{Total number of observation}}$ ,
Now as per the question it is given that the average of men is $32$ and the total number of men we have assumed is $x$ :
Therefore,

& \Rightarrow \text{Average}=\dfrac{\text{Sum of ages of men}}{\text{Total number of men}} \\\ & \Rightarrow \text{32}=\dfrac{\text{Sum of ages of men}}{x} \\\ & \Rightarrow 32x=\text{Sum of ages of men }...........\text{Equation 1}\text{.} \\\ \end{aligned}$$ Again we see that it is given that the average of women is $27$ and the total number of men we have assumed is $y$ : Therefore, $$\begin{aligned} & \Rightarrow \text{Average}=\dfrac{\text{Sum of ages of women}}{\text{Total number of women}} \\\ & \Rightarrow \text{27}=\dfrac{\text{Sum of ages of women}}{y} \\\ & \Rightarrow 27y=\text{Sum of ages of women }...........\text{Equation 2}\text{.} \\\ \end{aligned}$$ We now see that it is given that the average age of the combined group of men and women is $$30$$ years and the total number of men and women we will be $x+y$ : Therefore, $$\begin{aligned} & \Rightarrow \text{Average}=\dfrac{\text{Sum of ages of women and wom}}{\text{Total number of men and women}} \\\ & \Rightarrow \text{30}=\dfrac{\text{Sum of ages of women and men}}{x+y} \\\ & \Rightarrow 30\left( x+y \right)=\text{Sum of ages of women and men }...........\text{Equation 3}\text{.} \\\ \end{aligned}$$ Now we know from equation 1 that sum of ages of men is $32x$ and from equation 2 that sum of ages of women is $27y$ , therefore the total sum of ages of women and men is$32x+27y$ , putting this in the right hand side of equation 3: $$\begin{aligned} & \Rightarrow 30\left( x+y \right)=\text{Sum of ages of women and men} \\\ & \Rightarrow 30\left( x+y \right)=32x+27y\Rightarrow 30x+30y=32x+27y \\\ & \Rightarrow 30y-27y=32x-30x\Rightarrow 3y=2x \\\ & \Rightarrow \dfrac{x}{y}=\dfrac{3}{2} \\\ \end{aligned}$$ Therefore, $x:y=3:2$ We know that if a ratio is given $m:n$ and we have to find the percentage of $n$ then$n\%=\left( \dfrac{n}{m+n} \right)\times 100$ Now, $x:y=3:2$, where $x$ is the number of men and $y$ is the number of women and we have to find the percentage of women in the group so : $\begin{aligned} & y\%=\left( \dfrac{y}{x+y} \right)\times 100 \\\ & y\%=\left( \dfrac{2}{3+2} \right)\times 100\Rightarrow y\%=\left( \dfrac{2}{5} \right)\times 100 \\\ & \Rightarrow y\%=0.4\times 100\Rightarrow y\%=40 \\\ \end{aligned}$ Therefore the percentage of women in the group is $40\%$ . **Hence the correct option is B.** **Note** : Average is also known as the mean value that means it shows the most common value of a given item. The calculation is not tedious although when finding out the percentage from the ratio student needs to be a bit careful as we see that in $n\%=\left( \dfrac{n}{m+n} \right)\times 100$ sometimes student may forget to add $m$ and $n$ in the denominator from the ratio $m:n$ .