Let
c = number of leaps of cat
d = number of leaps of dog
$\dfrac{c}{d} = \dfrac{4}{3}$
$c = \dfrac{4d}{3}$
2 leaps of dog = 3 leaps of cat = L
$\text{1 leap of dog} = \dfrac{L}{2}$
$\text{1 leap of cat} = \dfrac{L}{3}$
For the dog to catch the cat:
$(c + 50)\left( \dfrac{L}{3} \right) = d \left( \dfrac{L}{2} \right)$
$\left(\dfrac{4d}{3} + 50 \right)\left( \dfrac{1}{3} \right) = d \left( \dfrac{1}{2} \right)$
$d = 300 ~ \text{leaps}$ ← answer