When sonometer is stretched by weight w (tension), then frequency of vibration in the string
$v=\frac{1}{2l}\sqrt{\frac{T}{m}}$
$v=\frac{1}{2L_1}\sqrt{\frac{w}{m}}$ … (i)
where, $L_1$ is the resonating length and m is mass per unit length of the string.
Similarly, when weight reduces to $\frac{w}{4}$ and
Resonating lengths is $L_2,$ then
$v=\frac{1}{2L_2}\sqrt{\frac{\frac{w}{4}}{m}}\Rightarrow v=\frac{1}{4L_2}\sqrt{\frac{w}{m}}$ …(ii)
From Eqs. (i) and (ii), we have
$\frac{1}{2L_1}\sqrt{\frac{w}{m}}\times \frac{1}{4L_2}\sqrt{\frac{w}{m}}\Rightarrow L_1=2L_2$
$\therefore \frac{L_1}{L_2}=2$
$\therefore L_1:L_2=2:1$