Please someone tell me the name of this algebra trick or where I can learn how to do it..

Question

Okay so my algebra knowledge is pretty guff..

I am taking a control systems class and pretty much all the questions I am expected to revise, are about doing this algebraic manipulation and I don't know what steps the tutor is taking to do it..

Okay here goes..

If the transfer function of a system is $G(s) = 3/(20s+1)$, then the closed loop version of that is

$$G(s)/(G(s) + 1)$$

so that would be

$$\frac{\frac{3}{20s+1}}{\frac{3}{20s+1} + 1}$$

This is the bit I am having trouble with.. he just then cancels it all out and gives us the answer on the next line which is.. $$\frac{3}{20s+4}$$

He gives us loads of problems to this which are all similar but I just cannot work them out as my algebra sucks so bad.. I don't know how to cancel out stuff which has a division but with an addition in the denominator..

I have taken a screen shot of the pdf here https://i.stack.imgur.com/TKpxh.png

And another one of another pdf which explains nearly how to do it but misses out the steps..

Answer 1

Note that $$\frac {\frac 3{20s+1}}{\frac3{20s+1}+1}=\frac {\frac 3{20s+1}}{\frac3{20s+1}+1}\cdot\frac {20s+1}{20s+1}=\frac{3}{3+20s+1}=\frac 3{20s+4}$$

Answer 2

\begin{align} \frac{\frac{3}{20s+1}}{\frac{3}{20s+1}+1} &= \frac{{20s+1}}{{20s+1}}\frac{\frac{3}{20s+1}}{\frac{3}{20s+1}+1}\\ & = \frac{\frac{3(20s+1)}{20s+1}}{\frac{3(20s+1)}{20s+1}+(20s+1)}\\ & = \frac{3}{3+(20s+1)}. \end{align}

Answer 3

Multiply nominator and denominator by $20s+1$.

Answer 4

You can usually resort to doing everything straightforwardly if you don't see how the "tricks" work.

In this case, the straightforward thing is to "compute" the sum:

$$ \frac{3}{20s + 1} + 1 = \frac{20s + 4}{20s+1} $$

There are lots of ways to see how to do this. You can try to find the least common denominator, but it's good enough to find any common denominator. i.e. you can write

$$ \frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{bc}{bd} = \frac{ad+bc}{bd} $$

Once you've done that, the next straightforward thing to do is to divide fractions. There are again several mnemonics for remembering this; a common one is

$$ \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc} $$