<s>[INST] How to simplify $\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$? $$\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$$
I have been staring at it for ages and know that it simplifies to $x$, but have been unable to make any significant progress.
I have tried doing $(\frac{1-x}{1-2x})(\frac{1+2x}{1+2x})$ but that doesn't help as it leaves $\frac{1+x-2x^2}{1-4x^2}$ any ideas? 
 [/INST] 
$$\frac{1 - \frac{1-x}{1 - 2x}}{1 - 2\frac{1-x}{1-2x}}$$

Multiplying through by $1-2x$ gives
$$\frac{1 - 2x - (1 - x)}{1 - 2x - 2(1 - x)}.$$
Expanding the brackets and then simplifying gives
$$\frac{1 - 2x - 1 + x}{1 - 2x - 2 + 2x}$$
$$= \frac{-x}{-1} = x.$$
NOTE: This method works well because you have the same "denominator" of $1 - 2x$ every where. If you had different ones, I  would recommend the finding the lowest common denominator method that is your accepted answer.
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