# Finding all the ordered pairs

How to find all the distinct positive integer ordered pairs for $a+b\le100$ such that :

$$\frac{a+\frac{1}{b}}{b+\frac{1}{a}}=10$$

• Simplify the LHS. :) – darij grinberg Aug 12 '13 at 22:59

Hint: $$\frac{a+\frac{1}{b}}{b+\frac{1}{a}}=\frac{a+\frac{1}{b}}{\frac{b}{a}(a+\frac{1}{b})}$$
• or multiplying top and bottom by $ab (\gt 0 )$ to give $\dfrac{a(ab+1)}{b(ab+1)} = 10$ – Henry Aug 12 '13 at 23:23
• @Henry Sure. ${}{}{}{}$ – Amr Aug 12 '13 at 23:45