the average incomes of the people in two villages are P and Q respectively .Assume that `pne Q .` A person moves form the first village to the second to the second village . The new average income are P and Q respectively which of the following is not possible ?

Options

A. `Pgt p and Q gt Q `
B. `pgt P and Qgt Q `
C. `P=P and Q =Q `
D. `P lt P and Q ltQ `

Answer

C

Explanation

`(x_(1) +x_(2) +.. . . .+ X_(n) )/(n) =p`
`(y_(1) +y_(2) +.. . . .+ y_(m) )/(m) =Q `
`implies x_(1) +x^(2) +.. .. .. + x_(n)=nP `
`& y_(1) + y_(2) +.. .. .. + y_(m) =mQ`
Now if a person moves from 1st village to II nd village then
`(x_(1) +x_(2) +.. .. +x_(n-1))/((n-1))=p'`
`implies nP -x_(n) =(n-1)P'implies p'=((n)/(n-1))p-((n)/(n-1))`
`and (y_(1) +y_(2) +.. .. + y_(m )+X_(n))/(M+1)=Q'`
` implies mQ+x_(n)=(m+1) Q'`
`therefore If p'=Pimplies p(1-(n)/(n-1))=(x_(n))/(n-1)`
`implies (-p) /(n-1)=-(x_(n))/(n-1)implies p=x_(n)`
& when Q=Q '
`implies x_(n) =Q=Q`
in that case p=Q (which is not true )
`therefore ` (c ) p'=P and Q =Q is not possible