\(a\frac{1}{b}:b\frac{1}{a}=\frac{ab+1}{b}:\frac{ab+1}{a}=\frac{ab+1}{b}\cdot\frac{a}{ab+1}=\frac{a}{b}\)
\(a\frac{1}{b}=\frac{a+b}{b}\)
\(b\frac{1}{a}=\frac{b+a}{a}\)
\(a\frac{1}{b}:b\frac{1}{a}=\frac{a+b}{b}:\frac{b+a}{a}=\frac{a+b}{b}x\frac{a}{a+b}=\frac{a}{b}\)