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 $1001$ dog owners, how many cat owners are there?

Let the number of cat owners be $C$.

Given, $70\%$ of cat owners also own a dog.

So, number of people who own both cat and dog:

$\frac{70}{100}C=\frac{7C}{10}$

Also, $20\%$ of dog owners also own a cat.

Number of dog owners is $1001$.

So, number of people who own both cat and dog:

$\frac{20}{100}\times 1001$

$=\frac{1}{5}\times 1001$

$=\frac{1001}{5}$

Now,

$\frac{7C}{10}=\frac{1001}{5}$

Multiply both sides by $10$:

$7C=2002$

$C=\frac{2002}{7}$

$C=286$