When we cut the spring into ratio of length $1: 2: 3,$ we

get three springs of lengths $\frac{l}{6}, \frac{2 l}{6}$ and $\frac{3 l}{6}$ with force

constant,

$\therefore k_{1}=\frac{k l}{l_{1}}=\frac{k l}{l / 6}=6 k$

${k_{2}=\frac{k l}{l_{2}}=\frac{k l}{2 l / 6}=3 k}$

${k_{3}=\frac{k l}{l_{3}}=\frac{k l}{3 l / 6}=2 k}$

When connected in series,

$\frac{1}{k^{\prime}}=\frac{1}{6 k}+\frac{1}{3 k}+\frac{1}{2 k}=\frac{1+2+3}{6 k}=\frac{1}{k}$

$\therefore \quad \overline{k^{\prime}}=k$

When connected in parallel,

${k^{\prime \prime}=6 k+3 k+2 k=11 k}$

${\frac{k^{\prime}}{k^{\prime \prime}}=\frac{k}{11 k}=\frac{1}{11}}$