t

_{1} = time for the sound of gun to reach the observer

$t_1 = \dfrac{1342 - x}{335}$

$t_1 = \dfrac{1342}{335} - \dfrac{x}{335}$

t_{2} = time for the sound of impact to reach the observer

$t_2 = \dfrac{1342}{503} + \dfrac{x}{335}$

$t_1 - \frac{1}{4} = t_2$

$\left( \dfrac{1342}{335} - \dfrac{x}{335} \right) - \dfrac{1}{4} = \dfrac{1342}{503} + \dfrac{x}{335}$

$\dfrac{2}{335}x = 1.088$

$x = 182.24 ~ \text{m from the target}$ *answer*