Gọi 3 phân số nhỏ nhất cần tìm là:\(\dfrac{a}{b};\dfrac{c}{d};\dfrac{m}{n}\left(b,d,n\ne0\right)\)
Ta có:
\(\dfrac{a}{b}+\dfrac{c}{d}+\dfrac{m}{n}=1\dfrac{1}{70}=\dfrac{71}{70}\)
\(\dfrac{a}{3}=\dfrac{c}{4}=\dfrac{m}{5}=h\\ \Rightarrow a=3h;c=4h;m=5h\\ \dfrac{b}{5}=\dfrac{d}{1}=\dfrac{n}{2}=k\\\Rightarrow b=5k;d=k;n=2k\)
\(\dfrac{3h}{5k}+\dfrac{4h}{k}+\dfrac{5h}{2k}=\dfrac{71}{70}\\ \dfrac{6h}{10k}+\dfrac{40h}{10k}+\dfrac{25h}{10k}=\dfrac{71}{70}\\ \dfrac{71h}{10k}=\dfrac{71}{70}\\ \Rightarrow h=1;k=7\\ \Rightarrow\left\{{}\begin{matrix}\dfrac{a}{b}=\dfrac{3h}{5k}=\dfrac{3}{35}\\\dfrac{c}{d}=\dfrac{4h}{k}=\dfrac{4}{7}\\\dfrac{m}{n}=\dfrac{5h}{2k}=\dfrac{5}{14}\end{matrix}\right.\)
Vậy ba phân số cần tìm là \(\dfrac{3}{35};\dfrac{4}{7};\dfrac{5}{14}\)
