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?
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The "intersection" of two sets is the key link in percentage-based word problems.
Step 1: Spot the shared quantity. The number of people who own both a cat and a dog can be counted from the dog side or the cat side; it is the same group either way. Step 2: Note the two percentages. 20 percent of dog owners also own a cat, and 70 percent of cat owners also own a dog. Step 3: Count both-owners from the dog side. With 1001 dog owners, the both-group $= 0.20 \times 1001 = 200.2$. Step 4: Express both-owners from the cat side. If $C$ is the number of cat owners, then the both-group $= 0.70 \times C$. Step 5: Set the two expressions equal. $0.70 \times C = 200.2$. Step 6: Solve for C. $C = \dfrac{200.2}{0.70} = 286$. So there are 286 cat owners. \[ \boxed{286} \]