If the plate rotates by seven complete revolutions of itself, equal to 5 circuits around the tray, it means that on each circuit the plate rotates by 7/5 of a revolution.

The centre of the circle traces the locus which is continuous and represented as the blue line.

As the points of an arc AB on the circumference of the circle come into contact with the tray, the centre of the circle moves along a path of length AB.

The distance the centre travels during 7 revolutions is 7 times $2\pi$ units.

As the disc completes 5 revolutions the total arc length of the rim that comes into contact with the tray in one revolution is ${14\pi \over 5}$. So $2p+2q= {14\pi \over 5}$, that is $p+q= {7\pi \over 5}$.

Any values of $p$ and $q$ which satisfy this equation will provide possible solutions.

The dimensions of the tray are $p+2$ and $q+2$ and there are many possible values of $p$ and $q$, for example $p =\pi$, $q={2\pi\over 5}$ or $p = q = {7\pi\over 10}$.