# 09 - Number of Leaps to Take to Catch the Lead

**Problem**

A cat takes 4 leaps to a dog's 3; but 2 of the dog's leaps are equivalent to 3 of the cat’s. The cat has a start of 50 leaps. How many leaps must the dog take to catch the cat?

**Answer Key**

300 leaps

**Solution**

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*

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