What dilutions would have to me made (from a starting concentration of 5 x 10^8 cells per ml) to achieve a final concentration of 1 x 10^3 cells per ml?
This is an inverse proportion problem:
5*10^8*1 = 1*10^3*V
V=5*10^8 / (1*10^3)
=5*10^5 times the initial volume.
So the dilution is
1 ml of the initial solution and
5*10^5-1 = 499999 litres diluent (water?)
to make a total volume of 500000 ml, or 500 litres of diluted "solution".