Simplifying complex fraction

Question

Trying to reduce this complex fraction to the answer given in my lecture notes. I've tried a few methods, which I haven't included here because it gets messy! Just wondering is there a trick/shortcut that I'm missing? I've tried reducing the small fractions in the big fraction and then flipping the denominator and multiplying it with the numerator.

$${\left({0.2(1-0.9Z^{-1}) \over 1-Z^{-1}}\right) \left({0.1 \over Z-0.9}\right) \over 1+ \left({0.2(1-0.9Z^{-1}) \over 1 - Z^{-1}}\right) \left({0.1 \over Z-0.9}\right)}={0.2Z^{-1} \over 10-9.8Z^{-1}}$$

Answer 1

Multiply numerator and denominator by $$\left(1-Z^{-1}\right)(Z-.9)$$ to get $$\frac{.02\left(1-.9Z^{-1}\right)}{\left(1-Z^{-1}\right)(Z-.9)+.02\left(1-.9Z^{-1}\right)}$$ Multiply numerator and denominator by $Z$ to get $$\frac{.02(Z-.9)}{(Z-1)(Z-.9)+.02(Z-.9)}$$ Now cancel $Z-.9$ from numerator and denominator, and you'll get an expression easily manipulated into the desired form.

I wonder why your instructor didn't use the form $$\frac1{50Z-49}$$