Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y = mx.$ The possible values of $m$ lie in an interval of length $\dfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a + b\,?$

Koda is making birdhouses for a park. the instructions tell him how many of each wooden shape is needed for one birdhouse. complete each statement to adjust the number of shapes for each new situation if he follows the given instructions. to make 8 birdhouses, koda will trapezoids.  for koda to make birdhouses, he will need 12 rectangles. if koda uses 4 squares to make birdhouses, he will use triangles.