97. Find the proper fraction that will increase by 3 times if its numerator is cubed, and 3 is added to the denominator.
Solution. Let $\frac{a}{b}$ be a proper fraction (where $a$ and $b$ are natural numbers, $a<b$), such that $\frac{a^{3}}{b+3}=3 \cdot \frac{a}{b}$, from which $\frac{a^{2}}{b+3}=\frac{3}{b}$, $a^{2} b=3 b+9, b\left(a^{2}-3\right)=9$. From this, it is clear that the number $b$ must be a divisor of the number 9, i.e., $b$ is either 1, 3, or 9. Let's check these values:

1) if $b=1$, then $a=\sqrt{12}$;
2) if $b=3$, then $a=\sqrt{6}$;
3) if $b=9$, then $a=2$.

Since $a$ is a natural number, the first two cases do not fit.

Answer: $\frac{2}{9}$.