Answer: 200

Solution:

Write $r = \frac{a}{b}$ in lowest terms with $a + b = 1000;$ since $0 \lt r \lt 1,$ we need $1 \le a \le 499.$ Because $\gcd(a, b) = \gcd(a, 1000 - a)$ $= \gcd(a, 1000),$ the fraction is in lowest terms exactly when $a$ is coprime to $1000.$

There are $\varphi(1000) = 1000 \cdot \frac{1}{2} \cdot \frac{4}{5} = 400$ integers in $[1, 999]$ coprime to $1000,$ and they pair up as $a \leftrightarrow 1000 - a$ (note $a = 500$ is not coprime to $1000$), so exactly $200$ of them are less than $500.$ The answer is $200.$