The length of sonometer wire between two fixed ends is 100 cm. Three bridges be placed so as to divide the wire into four segments whose fundamental frequencies are in the ratio of 1 : 2 : 3 : 4 from the left end. Then which of the following is/are correct.

 ${\ell }_1+{\ell }_2+{\ell }_3+{\ell }_4=100cm$
Let  $n_1,n_2,n_3$  and $n_4$  are fundamental frequencies of these segments respectively.
 
 
$\frac{n_1}{n_2}=\frac{1}{2},\frac{n_2}{n_3}=\frac{2}{3},\frac{n_3}{n_4}=\frac{3}{4},\frac{n_2}{n_4}=\frac{2}{4}=\frac{1}{2}$      $n=\frac{1}{2\ell }\sqrt{\frac{T}{\mu }}\Rightarrow n\ell$  = constant
 
 $n_1{\ell }_1=n_2{\ell }_2=n_3{\ell }_3=n_4{\ell }_4$ 

$$\begin{gathered} {\ell }_1=\frac{n_2}{n_1}{\ell }_2,{\ell }_3=\frac{n_2}{n_3}{\ell }_2,{\ell }_4=\frac{n_2}{n_4}{\ell }_2 \\ {\ell }_1+{\ell }_2+{\ell }_3+{\ell }_4=100cm \\ 2{\ell }_1+{\ell }_2+\frac{2}{3}{\ell }_2+\frac{{\ell }_2}{2}=100 \\ {\ell }_2=24cm \end{gathered}$$
 
 So, ${\ell }_1$  = 48 cm

$$\begin{gathered} {\ell }_3=16cm \\ {\ell }_4=12cm \end{gathered}$$