If `x_1,x_2,.......,x_n` are n values of a variable `X` such that `sum_(i=1)^n (x_i-2)=110` and `sum_(i=1)^n(x_i-5) =20`. Find the value of `n` and the mean.

Explanation

Solution:
`sum x i=x_1+x_2+...+x_n`
So, `sum(x i - 2)=(x_1−2)+(x_2 −2)+...+(x_n−2)`
Also, `sum( x i- 2)=110`
`=>(xi−2)+(x_2 −2)+...+(xn −2)=110`
`=>(x_1 +x_2​+..+x_n )−2n=110`
`=>sum x i−2n=110` ----`(1)`
Similarly,
`sum (x i −5)=(x_1 −5)+(x_2−5)+...+(x_n −5)`
Also, `sum(xi) −5)=20`
`=>(x_1 −5)+(x_2 −5)+...+(x_n−5)=20`
`=> (x_1+x_2 +...+x_n)−5n=20`
`=>sum x i−5n=20` ----`(2)`
Subtracting `(1)` from `(2)` we get,
`=>sum x i)−5n−(sum x i-2n)= 20−110`
`=−3n=−90`
`:. n=30`
`=>sum(x i)−5(30)=20`
`=>sum x i​ =20+150`
`:.sum x i=170`
`therefore` Mean`=sum (x i )/n)=170/30=5.67`