There are two A.P.'s each of $n$ terms $a,a+d,a+2d,...L$,  $p,p+q,p+2q,....L^{\prime }$. These A.P.'s satisfy the following conditions: $\frac{L}{p}=\frac{L^{\prime }}{a}=4,\frac{S_n}{S_{n^{\prime }}}=2;$ find out 2($\frac{L}{L^{\prime }}$)

$\frac{L}{p}=\frac{L^{\prime }}{a}=4$  (given)...(1)

$a+\left(n-1\right)d=4p$

$\frac{4p-a}{4a-p}=\frac{d}{q}$...(2)

$\frac{S_n}{{S_n}^{\prime }}=\frac{\frac{n}{2}\left(a+L\right)}{\frac{n}{2}\left(p+L^{\prime }\right)}=\frac{a+4p}{p+4a}=2$  given  by  (1)

$2p=7aor\frac{a}{2}=\frac{p}{7}=\lambda$  ...(3)

Putting  in  (2)  

$\frac{d}{q}=\frac{28\lambda -2\lambda }{8\lambda -7\lambda }=26$  and  

$\frac{L}{L^{\prime }}=\frac{p}{a}=\frac{7}{2}$  by  (1)  and  (3)

$2\frac{L}{L^{\prime }}=7$