A cone is cut half way through its axis and parallel to the base, the volumes of two portions are in the ratio?

let vertex of cone is  A ...

after drawing the axis about which cone is to cut  , then we will see that there will be two similar triangles  ...

radius of base of upper half is r¹ & radius of base is R , height of original cone is 2H then

by using similarity of triangles , r¹/H = R/2H

r¹ = R/2     

area of upper half = 1/3 (pi) (r¹)²H = 1/3(pi)(R²H/4)

area of lower half = area of cone - area of upper half

                         =1/3(pi)(R²2H) - 1/3(pi)(R²H/4)

                         =1/3pi(7R²H/4)

ratio of two parts = upper half/lower half = 1/7