Three long straight wires are connected parallel to each other across a battery of negligible internal resistance. The ratio of their resistances are 3 : 4 : 5. What is the ratio of distances of middle wire from the others if the net force experienced by it is zero

The wires are in parallel, and ratio of their resistances are $3:4:5$, Hence currents in wires are in the ratio

$i_1=\frac{k}{3}$, $i_2=\frac{k}{4},$$i_3=\frac{k}{5}$
 
Force per unit length between top and middle wire:

$F_1=\frac{{\mu }_0}{4\pi }\cdot \frac{2i_1{i}_2}{r_1}$$=\frac{{\mu }_0}{4\pi }\times \frac{2\left(\frac{1}{3}\right)\left(\frac{1}{4}\right)k^2}{r_1}$

similarly, Force between bottom and middle wire:

$F_2=\frac{{\mu }_0}{4\pi }\frac{{\displaystyle 2}\left(\frac{1}{4}\right)\left(\frac{1}{5}\right)k^2}{r_2}$

As the forces are equal and opposite, 

$\Rightarrow F_1=F_2$

$\Rightarrow \frac{r_1}{r_2}=\frac{5}{3}$