Hi,
please help me understand how this fraction:
b/[b-(a-1)]
can be rearranged to form:
(b/b+1)/{1-[1-b/(b+1)]a}
I would really appreciate it.
Well, it's a bit tricky, but here we go.
I see a b+1 as a denominator on both the top and the bottom, so I'll start by dividing both sides by b+1 to see what happens:
$\displaystyle $\displaystyle \frac{b}{b-(a-1)} = \frac{\frac{b}{b+1}}{\frac{b+1-a}{b+1}} = \frac{\frac{b}{b+1}}{\frac{b+1}{b+1}-\frac{a}{b+1}} = \frac{\frac{b}{b+1}}{1-\frac{a}{b+1}} $$
Note that on the first step i've expanded the brackets and rearanged it to b+1-a. You can see that we are almost there.
$\displaystyle $\displaystyle \frac{\frac{b}{b+1}}{1-\frac{a}{b+1}} = \frac{\frac{b}{b+1}}{1-\frac{a(b+1-b)}{b+1}} = \frac{\frac{b}{b+1}}{1-\frac{a(b+1)}{b+1}+\frac{ab}{b+1}} = \frac{\frac{b}{b+1}}{1-a+\frac{ab}{b+1}} = \frac{\frac{b}{b+1}}{1-a(1-\frac{b}{b+1})}$$
Which is what you needed. This is probably how you would write it. Here are some tips on how to do these things. First, if you have to make one simple expression into a more complicated expression, just start with the more complicated one and put stuff together (fractions for example, and roots). Make it as simple as you can. If you get it this way, then, you can either leave it at that or you can just write all the steps in reverse like I did. I actually wrote it the other way around.