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? 

When four storey fractions are involved, it's better to do computations separately. Write
$$
N=1-\frac{1-x}{1-2x},\qquad
D=1-2\frac{1-x}{1-2x}
$$
and work on $N$ and $D$:
\begin{align}
N&=1-\frac{1-x}{1-2x}\\
&=\frac{(1-2x)-(1-x)}{1-2x}\\
&=\frac{1-2x-1+x}{1-2x}\\
&=\frac{-x}{1-2x}
\end{align}
and
\begin{align}
D&=1-2\frac{1-x}{1-2x}\\
&=\frac{(1-2x)-2(1-x)}{1-2x}\\
&=\frac{1-2x-2+2x}{1-2x}\\
&=\frac{-1}{1-2x}
\end{align}
Now you know that your fraction is
$$
\frac{N}{D}=N\cdot\frac{1}{D}=\frac{-x}{1-2x}\cdot\frac{1-2x}{-1}
=\frac{-x}{-1}=x
$$
