The tangent galvanometers having coils of the same radius are connected in series. A current flowing in them produces deflections of ${60}^0$& ${45}^0$ respectively. The ratio of the number of turns in the coil is

In the first galvanometer  $i_1=K_1\tan {\theta }_1$
                                                   $i_1=K_1\tan {60}^0=\sqrt{3}{K}_1$
In the second galvanometer   $i_2=K_2\tan {\theta }_2$
                                                $i_2=K_2\tan {45}^0=K_2$
In series combination  $i_1=i_2$
 $$\begin{gathered} \therefore \sqrt{3}{K}_1=K_2 \\ \Rightarrow \frac{K_1}{K_2}=\frac{1}{\sqrt{3}} \end{gathered}$$
But  $K\propto \frac{1}{n}$
 $\Rightarrow \frac{K_1}{K_2}=\frac{n_2}{n_1}$
$\Rightarrow \frac{n_1}{n_2}=\frac{K_2}{K_1}=\frac{\sqrt{3}}{1}$