Answer
$\frac{v_S}{v} = \frac{1}{3}$
Work Step by Step
If the wavelength of the sound detected at A is less than the wavelength of the sound detected at B, the source must be moving in the direction from B to A.
We can find an expression for the frequency detected at A:
$f_A = f~\frac{v}{v-v_S}$
We can find an expression for the frequency detected at B:
$f_B = f~\frac{v}{v+v_S}$
If the wavelength of the sound detected at A is $0.500$ that of the sound detected at B, then $f_A = 2~f_B$
We can find the ratio $\frac{v_S}{v}$:
$f_A = 2~f_B$
$f~\frac{v}{v-v_S} = 2~ f~\frac{v}{v+v_S}$
$v+v_S = 2v-2v_S$
$3~v_S = v$
$\frac{v_S}{v} = \frac{1}{3}$
