Question 1088524
Compared to the original tetrahedron, the smaller tetrahedron cut off is a similar shape, meaning a scaled-down version.
For similar shapes, the ratios of lengths (L and l), surface areas (A and a), and volumes (V and v) are related by
{{{v/V=(l/L)^3}}} and {{{a/A=(l/L)^2}}} .
In this case, if {{{x}}} is the fraction of the altitude of a tetrahedron cut off by a plane parallel to the base of the tetrahedron,
and the volume of the small cut off tetrahedron is one third of the original volume,
{{{1/3=x^3}}} --> {{{x=root(3,1/3)=root(3,9/27)=root(3,9)/3=about0.693}}} .
The exact result is an irrational number, so it cannot be expressed as a fraction.
However, the fractions {{{52/75}}} and {{{658/949}}} are good approximate values.