Problem 9.3. The teacher wrote a fraction on the board, where the numerator and the denominator are natural numbers. Misha added 30 to the numerator of the given fraction and wrote the resulting fraction in his notebook, while Lesha subtracted 6 from the denominator of the fraction written on the board and also wrote the resulting fraction in his notebook. The fractions recorded by the boys turned out to be equal to the same number. What is this number?
Answer: 5.

Solution. Let $\frac{a}{b}$ be the original fraction. Then Misha wrote down the fraction $\frac{a+30}{b}$ in his notebook, and Lёsha wrote down $-\frac{a}{b-6}$.

Let's write the equation

$$
\frac{a+30}{b}=\frac{a}{b-6}
$$

Transforming it, we get

$$
\begin{gathered}
(a+30)(b-6)=a b \\
a b+30 b-6 a-180=a b ; \\
30 b=6 a+180 \\
\frac{a+30}{b}=5
\end{gathered}
$$

This is the value of the fraction obtained by Misha. It should also have been obtained by Lёsha.