Two springs A and B have force constants $k_1$ and $k_2$ respectively. The ratio of the work done on A to that done on B when they are stretched by the same force is

When the springs are stretched by the same force F, the extensions in springs A and B are $x_1$ and $x_2$ respectively which are given by
$F=k_1{x}_1=k_2{x}_2 \text{ or } \frac{x_1}{x_2}=\frac{k_2}{k_1}$………(1)
Work done $W_1=\frac{1}{2}{k}_1{x}_1^2$ and $W_2=\frac{1}{2}{k}_2{x}_2^2$
$\therefore \frac{W_1}{W_2}=\frac{k_1}{k_2}·\frac{x_1^2}{x_2^2}$…………..(2)
Using (i) and (ii) we get
$\frac{W_1}{W_2}=\frac{k_1}{k_2}·\frac{k_2^2}{k_1^2}=\frac{k_2}{k_1}$