To what fraction of its current radius would the sun have to be compressed to create a black hole?

You want to determine the radius of the sphere that the
present volume of the sun would have to be compressed in, to achieve a black
hole.

We can determine this using a formula developed by
the mathematician named Karl Schwarzschild.

According to
the formula, to create a black hole starting with a body of mass M, the radius of the
sphere it has to be compressed to is given by Rs = 2*G*M/c^2, where G is the
gravitational constant, M is the mass of the body we want to compress and c is the speed
of light.

The mass of the Sun is 1.99*10^30 kg, G=
6.674*10*10^-11 N*(m/kg) ^2 and the speed of light is 299792458
m/s.

Now using these values in the formula, we get Rs =
2*1.99*10^30*6.674*10^-11/ (299792458) ^2

= 2955
m

The present radius of the Sun is 7*10^5 km = 7*10^8
m.

Therefore we need to compress this into a sphere with a
radius that is 2955/7*10^8 = 4.2*10^-6 times the present
radius.

The ratio of the radius of the sphere
required to achieve a black hole to the present radius of the Sun is equal to
4.2*10^-6.