Thread: complex fractions

1. complex fractions

what is the technique when solving complex fractions?

a
--------
1 - 1
----
1+ 1
---
a-1

2. invert and multiply

3. Fractions

Hello 21_knip
Originally Posted by 21_knip
what is the technique when solving complex fractions?

a
--------
1 - 1
----
1+ 1
---
a-1
If I read your attachment correctly, start with the 1 and the fraction at the bottom first, and write them over a common denominator; like this:

$\displaystyle 1 + \frac{1}{a-1} = \frac{a-1}{a-1}+\frac{1}{a-1}$

$\displaystyle = \frac{a-1+1}{a-1}$

$\displaystyle = \frac{a}{a-1}$

You now need 1 over this answer; that's $\displaystyle \frac{1}{\frac{a}{a-1}}$

This is simply the reciprocal of the fraction: turn it 'upside-down': $\displaystyle \frac{a-1}{a}$

Now you do a similar thing for the remaining 1 and this new fraction:

$\displaystyle 1 - \frac{a-1}{a} = \frac{a}{a} - \frac{a-1}{a}$

$\displaystyle = \frac{a-a{\color{red}+}1}{a}$ Watch that sign!

$\displaystyle = \frac{1}{a}$

And so finally (as chiph588@ said) invert and mulitply:

$\displaystyle \frac{a}{\frac{1}{a}}$

$\displaystyle = a \times \frac{a}{1}$

$\displaystyle = a^2$

I hope you followed all that.

Grandad