1. ## Fractions

$\frac{x}{1-\frac{1}{1-x}}$

Thank you!

2. Straightforward

$
\mathcal A=\frac x{1-\dfrac1{1-x}}=\frac{1-x}{1-x}\cdot\frac x{1-\dfrac1{1-x}}$

So we have


\begin{aligned}\mathcal A&=\frac{x(1-x)}{(1-x)-1}\\
&=\frac{x(1-x)}{-x}\\
&=\frac{x(x-1)}x\\
&=x-1
\end{aligned}

3. Thank you!
But I don't understand, why do we multiply by $1-x$ ?

Because I almost got to the same conclusion by using this method

$X:1-\frac{1}{1-x}$

then I multiplied $1*(1-x)$ to make the denominators the same.

Then I followed the standard procedure for dividing fractions, in other words I multiplied x by the reciprocal $\frac{1-x}{-x}$

and then I also got $\frac{x(1-x)}{x}$

but then I don't understand, how one gets rid of the minus below?

4. Originally Posted by Coach
Thank you!
But I don't understand, why do we multiply by $1-x$ ?

Because I almost got to the same conclusion by using this method

$X:1-\frac{1}{1-x}$

then I multiplied $1*(1-x)$ to make the denominators the same.

Then I followed the standard procedure for dividing fractions, in other words I multiplied x by the reciprocal $\frac{1-x}{-x}$

and then I also got $\frac{x(1-x)}{x}$

but then I don't understand, how one gets rid of the minus below?
Krizalid just applied a nice trick to clear the fractions in the denominator without actually combining them.

note that he removed the minus in the denominator by changing the (1 - x) in the top to (x - 1)

5. Originally Posted by Jhevon
Krizalid just applied a nice trick to clear the fractions in the denominator without actually combining them.

note that he removed the minus in the denominator by changing the (1 - x) in the top to (x - 1)

Thank you!

Does this method have a name so I can google it and learn it for further use?

6. Originally Posted by Coach
Thank you!

Does this method have a name so I can google it and learn it for further use?
i don't think so. the trick is to just multiply the top and bottom of the whole fraction by the common denominator of the fractions in the denominator of the big fraction

7. Originally Posted by Coach
Thank you!

Does this method have a name so I can google it and learn it for further use?
Look up "complex fractions."

-Dan