$\frac{35*87}{67*59}$ = $\frac{30*18.\overline{45}}{(39*18.\overline{45}) - 1}$
edited version:
Step by step, how would I prove that the left and right sides are exactly equal using fractional math only (no decimal math)?
$\frac{35*87}{67*59}$ = $\frac{30*18\frac{45}{99}}{(39*18\frac{45}{99}) - 1}$
By the usual technique for converting repeating decimals to fractions, prove that $$0.\overline {45}=\frac {45}{99}$$ Now plug that into your equation.
Added: now use $18\frac {45}{99}=\frac {1827}{99}$ because mixed fractions are a pain. So $$\frac{30*18\frac{45}{99}}{(39*18\frac{45}{99}) - 1}=\frac {30 \cdot\frac {1827}{99}}{39\cdot \frac{1827}{99}-1}\\ =\frac{30\cdot 1827}{39\cdot 1827-99}\\ =\frac{35\cdot 87 \cdot 18}{71154}\\= \frac{35\cdot 87 \cdot 18}{67\cdot59\cdot18}\\ =\frac{35\cdot 87}{67\cdot 59}$$
