Start by considering the center of the spheres, which is also the geometric center of the tetrahedron, which we will call T. From this center, draw a ray to each vertex of tetrahedron T. These rays form the edges of four smaller, irregular tetrahedrons, t1 thru t4, each with the center point of T at its apex and a side of T as its base. From symmetry, t1 thru t4 are dimensionally identical.
In general, the volume of a tetrahedron is 1/3*base*height. Since t1 is 1/4 the volume of T (t1 thru t4 add up to T)and has the same base, the height of t1 must be 1/4 the height of T. And since the height of t1 plus the length of a ray equal the height of T, the length of each ray must be 3/4 the height of T.
Since the height of t1 is the radius of the inner sphere, and the ray is the radius of the outer sphere, the ratio of radii for the two spheres is 1/4 to 3/4, or 1:3.