Question

The average age of persons in committee is increased by 2 years when two men aged 35 years and 45 years are substituted by two women. What is the average of these two women?
$\begin{aligned} & \text{a) 54 years} \\\ & \text{b) }\text{52 years} \\\ & \text{c) 50 years} \\\ & \text{d) 48 years} \\\ \end{aligned}$

Answer 1

Now first let us assume that the sum of ages of all members except two men is S. Now we can find the average age of the people in committee with the formula $\dfrac{\text{Sum of the values}}{\text{number of values}}$. Now we will replace two men with two women, let x be the sum of the ages of two women. Now again we will find the new average of age of people in the committee. Hence we can now use the condition that the new average is 2 more than the previous age and hence we get the value of x. Now we can easily find the average age of two women by formula $\dfrac{\text{Sum of the values}}{\text{number of values}}$ .

Complete step by step answer:
Now we know that the committee has 8 people. Among those 8 people the age of 2 men is 35 years and 45 years. Let the sum of age of rest 6 people be S.
Now let us find the sum of age of all 8 people.
Sum of ages of 8 people will be = S + 35 + 45.
Now we know that average is nothing but $\dfrac{\text{Sum of the values}}{\text{number of values}}$ .
Hence the average age in the committee will be $\dfrac{S+35+45}{8}=\dfrac{S+80}{8}..........................\left( 1 \right)$
Now again consider the committee of 8 people.
Now the sum of age of all the members except two men is S.
Now these two men are replaced by two women.
Now let the sum of ages of these two women be x.
Hence we have the sum of ages of the committee will be S + x.
Now again the formula to calculate the average of values is $\dfrac{\text{Sum of the values}}{\text{number of values}}$ .
Hence we get the new average is $\dfrac{S+x}{8}...............................\left( 2 \right)$
Now we are given that average age is increased by 2 years when two men are substituted by two women.
Hence from equation (1) and equation (2) we get.
$\begin{aligned} & \dfrac{S+80}{8}+2=\dfrac{S+x}{8} \\\ & \Rightarrow \dfrac{S+80+16}{8}=\dfrac{S+x}{8} \\\ \end{aligned}$
$\begin{aligned} & \Rightarrow S+96=S+x \\\ & \therefore x=96 \\\ \end{aligned}$
Hence the sum of ages of two women is 96.
Now we want to calculate the average age of two women.
Average age of two women will be $\dfrac{\text{sum of age of two women}}{2}$
Hence average age is $\dfrac{96}{2}=48$
Hence the average age of two women is 48.

Option d is the correct option.

Note:
Now while calculating the average note that we need to find both the averages first before applying the condition. We cannot directly consider just the two men, since other data are not changing at all.