Answer
If $f$ and $g$ are periodic with period $p$, then $\frac{f}{g}$ is also periodic but its period could be smaller than $p$
Work Step by Step
$f(x) = f(x+p)$ and $g(x) = g(x+p)$ by the definition of periodic functions.
Let's consider $\frac{f}{g}$:
$\frac{f(x)}{g(x)} = \frac{f(x+p)}{g(x+p)}$
We can see that $\frac{f}{g}$ is also periodic. The period could be $p$, or $p$ could be subdivided into two or more smaller periods.
To see an example, we can consider $f(x) = sin~x$ and $g(x) = cos~x$
$\frac{f(x)}{g(x)} = \frac{sin~x}{cos~x} = tan~x$
The period of both $sin~x$ and $cos~x$ is $2\pi$
However, the period of $tan~x$ is $\pi$
Therefore, if $f$ and $g$ are periodic with period $p$, then $\frac{f}{g}$ is also periodic but its period could be smaller than $p$