284. How great is the number that equals the product of $\frac{4}{5}$ of it by $\frac{5}{6}$ of the same number?

Problem from an algebraic Latin manuscript p. 80 (Dresden collection).
284. In the manuscript, it is said: let the number be $n$. Four fifths of it will be $\frac{4}{5} n$. Five sixths of it will be $\frac{5}{6} n$. The product of $\frac{4}{5} n$ and $\frac{5}{6} n$ equals the number, i.e., $\frac{2}{3} n^{2}=n$. According to the form of the equation $a x^{2}=b x$, we have

$$
1 \cdot n=\frac{2}{3} n^{2}
$$

from which the number is $1 \frac{1}{2}$.

285 The first part is 4, the second part is 6. Indeed

$$
\frac{4 \cdot 5}{6}=\frac{10}{3}
$$