Solution

The velocity of the transverse waves on string is given as
$$v = \sqrt \frac{T}{\mu}$$ ..............(1)
where, T is the tension in the string and $$\mu$$ is the mass per length.
$$\therefore \mu = \dfrac{\pi r^2 \rho l}{l}$$
$$\therefore \mu = \pi r^2 \rho $$
$$\therefore v = \sqrt \frac{T}{\pi r^2 \rho}$$ ..............from (1)
$$\therefore v \propto \dfrac{1}{r}$$
where,
r is the radius of string
$$\rho$$ is the material density for string
Now as per the given problem both strings are of same materials and same tension tension
Therefore,
$$\dfrac{v_A}{v_B} = \dfrac{r_B}{r_A}$$
$$\dfrac{v_A}{v_B} = \dfrac{r_B}{2 r_B}$$
$$\dfrac{v_A}{v_B} = \dfrac{1}{2}$$