13.028. The numerators of three fractions are proportional to the numbers $1,2,5$, and the denominators are proportional to the numbers $1,3,7$ respectively. The arithmetic mean of these fractions is 200/441. Find these fractions.
Solution.

The required fractions have the form: $\frac{x}{y} ; \frac{2 x}{3 y} ; \frac{5 x}{7 y}$. According to the condition

$\frac{\frac{x}{y}+\frac{2 x}{3 y}+\frac{5 x}{7 y}}{3}=\frac{200}{441}$, from which $\frac{x}{y}=\frac{4}{7} ; \frac{2 x}{3 y}=\frac{8}{21} ; \frac{5 x}{7 y}=\frac{20}{49}$.

Answer: $\frac{4}{7} ; \frac{8}{21} ; \frac{20}{49}$