Question

The average incomes of the people in two villages are $P$ and $Q,$ respectively. Assume that $P\ne Q.$ A person moves from the first village to the second village. The new average incomes are $P^{\prime }$ and $Q^{\prime },$ respectively. Which of the following is not possible ?

• A. $P^{\prime }>P\text{and}{Q}^{\prime }>Q$
• B. $P^{\prime }>P\text{and}{Q}^{\prime }<Q$
• C. $P^{\prime }=P\text{and}{Q}^{\prime }=Q$
• D. $P^{\prime }<P\text{and}{Q}^{\prime }<Q$
  

Answer 1

The correct option is C $P^{\prime }=P\text{and}{Q}^{\prime }=Q$

Average incomes of the people in first villages $P=\frac{x_1+x_2+....+x_n}{n}$ and average incomes of the people in second villages $Q=\frac{y_1+y_2+....+y_m}{m}$
$\Rightarrow x_1+x_2+....+x_n=nP\text{\&}{y}_1+y_2+....+y_m=mQ$
Now if a person moves from first village to second village then
$\frac{x_1+x_2+....+x_{n-1}}{n-1}=P^{\prime }\Rightarrow nP-x_n=(n-1)P^{\prime }$
$\Rightarrow P^{\prime }=\frac{n}{n-1}P-\frac{x_n}{n-1}$
and
$\frac{y_1+y_2+....+y_m+x_n}{m+1}=Q^{\prime }\Rightarrow mQ+x_n=(m+1)Q^{\prime }$
Now if $P^{\prime }=P\Rightarrow (1-\frac{n}{n-1})P=-\frac{x_n}{n-1}\Rightarrow P=x_n$
Also if $Q^{\prime }=Q\Rightarrow Q=x_n$
Hence, $P=Q$ which is not possible as $P\ne Q$