Question

A cone is divided into two part by drawing a plane through the mid point of its axis and // to its axis and // to its base compare the volumes or two part

Answer 1

after dividing ,upper part just like cone and lower part just like frustum .
now,
let radius of cone of upper part formed by division =x
use ,
R/x = H/H/2 =2

R = 2x

now , volume of upper cone (smaller )= 1/3πr²h
=1/3πx²H/2 = 1/6πx²H

volume of lower part = volume of big cone - volume of smaller cone

=1/3πR²H -1/3πr²h
=1/3π{(2x)²H -x²H/2}
=1/3π{ 7x²H}/2
=7/6πx²H

now , volume of upper /volume of lower =1/6πx²H/7/6πx²H = 1/7

Answer 2

Let r be radius of smaller cone
Let R be radius of bigger cone
See figure, We got right angled triangle: 
So, x/2x = r/R ⇒ 1/2 = r/R ⇒ R/r = 2/1
Then comparing  = volume of cone/ volume of frustum 
                            =   1/3πr²x ÷ 1/3πx(r²+R²+Rr)
                            =   1 ÷  r²/r² + R²/r² + Rr/r 
                            =   1 ÷ 1 + (R/r)² + R/r 
                            =   1  ÷ 1+2²+2 
                            =    1/7 or 1:7