Call three fractions is A,B,C

We have : \(A=\dfrac{x}{y}\) ; \(B=\dfrac{y}{t}\) ; \(C=\dfrac{e}{f}\)

Follows the threads : \(\dfrac{x}{3}=\dfrac{z}{4}=\dfrac{e}{5}\) ; \(\dfrac{y}{5}=\dfrac{t}{1}=\dfrac{f}{2}\)

\(\Rightarrow\dfrac{\dfrac{x}{3}}{\dfrac{y}{5}}=\dfrac{\dfrac{z}{4}}{\dfrac{t}{1}}=\dfrac{\dfrac{e}{5}}{\dfrac{f}{2}}\)

\(\Rightarrow\dfrac{x}{3}.\dfrac{5}{y}=\dfrac{z}{4}.\dfrac{1}{t}=\dfrac{e}{5}.\dfrac{2}{f}\)

\(\Rightarrow\dfrac{x}{y}.\dfrac{5}{3}=\dfrac{z}{t}.\dfrac{1}{4}=\dfrac{e}{f}.\dfrac{2}{5}\)

\(\Rightarrow A:\dfrac{3}{5}=B:4=C:\dfrac{5}{2}\)

Applying the same sequence properties we have:

\(\dfrac{A}{\dfrac{3}{5}}=\dfrac{B}{4}=\dfrac{C}{\dfrac{5}{2}}=\dfrac{A+B+C}{\dfrac{3}{5}+4+\dfrac{5}{2}}=\dfrac{\dfrac{213}{70}}{\dfrac{71}{10}}=\dfrac{3}{7}\)

=> \(\left\{{}\begin{matrix}A=\dfrac{3}{7}.\dfrac{3}{5}=\dfrac{9}{35}\\B=\dfrac{4.3}{7}=\dfrac{12}{7}\\C=\dfrac{5}{2}.\dfrac{3}{7}=\dfrac{15}{14}\end{matrix}\right.\)