A researcher found that 70% of people who own a cat also own a dog, but only 20% of dog owners also own a cat. If there are 1,001 dog owners, how many cat owners are there?
Step-by-step Solution:
Let \( D \) be the total number of dog owners and \( C \) be the total number of cat owners. We are given \( D = 1001 \). The number of people who own both a cat and a dog is given by two different percentage perspectives. From dog owners: 20% of dog owners own a cat. \[ \text{Intersection} = 0.20 \times 1001 = 200.2 \] From cat owners: 70% of cat owners own a dog. \[ \text{Intersection} = 0.70 \times C \] Setting these equal: \[ 0.70 \times C = 200.2 \] \[ C = \frac{200.2}{0.70} \] \[ C = 286 \] There are 286 cat owners.