A spring of force constant \( k \) is cut into lengths of ratio \( 1:2:3 \). They are connected in series and the new force constant is \( k' \). Then they are connected in parallel and force constant is \( k'' \). Then \( k' : k'' \) is

Spring constant \( k \propto \frac{1}{l} \). The parts have lengths \( \frac{l}{6}, \frac{2l}{6}, \frac{3l}{6} \), so their constants are \( 6k, 3k, 2k \). In series, \( \frac{1}{k'} = \frac{1}{6k} + \frac{1}{3k} + \frac{1}{2k} = \frac{1}{k} \) so \( k' = k \). In parallel, \( k'' = 6k + 3k + 2k = 11k \). Thus \( k' : k'' = 1:11 \).