Answer
$\frac{F'}{F} = \frac{3}{8}$
Work Step by Step
Let the initial charge on each sphere be $q$
We can write an expression for the original force between the two spheres:
$F = \frac{q^2}{4\pi~\epsilon_0~r^2}$
After sphere 3 touches sphere 1, both sphere 1 and sphere 3 have a charge $\frac{q}{2}$
After sphere 3 touches sphere 2, both sphere 2 and sphere 3 have a charge $\frac{\frac{q}{2}+q}{2} = \frac{3q}{4}$
We can find the new force between the two spheres:
$F' = \frac{(\frac{q}{2})(\frac{3q}{4})}{4\pi~\epsilon_0~r^2} = \frac{3}{8}~\frac{q^2}{4\pi~\epsilon_0~r^2}$
We can find $\frac{F'}{F}$:
$\frac{F'}{F} = \frac{\frac{3}{8}~\frac{q^2}{4\pi~\epsilon_0~r^2}}{\frac{q^2}{4\pi~\epsilon_0~r^2}} = \frac{3}{8}$
