If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, what is the ratio of the volumes of the upper part and the cone?

When the cone is cut like said in the question, let us suppose the radius of small cone is r and of original cone is R. Let us suppose their heights are h and H respectively.

 

Now, these two cones will be similar, hence r/R = h/H = 1/2

 

Now ratio of volume of small cone and big cone will be

 

?r²h/?R²H = (r²/R²)(h/H) = 1/8