# Thread: Given (a+b)(c+d) = ab+ac+bc+bd, How to express (a+b)÷(c+d) OR

1. ## Given (a+b)(c+d) = ab+ac+bc+bd, How to express (a+b)÷(c+d) OR

Problem: Generating an expression that removes the parentheses using first principles.

Given (a+b)(c+d) = ab+ac+bc+bd,

Can we say: (a+b)÷(c+d) = a/c+a/d+b/c+b/d is True? (no we can't if we use test data!)

What I really want is to express R, so far I know R = (360 over W) - 10 - (R.(T -75W) over W)
So I'm trying to isolate for R on LHS only.

Workings:
If a=b=1 and c=d=2 then
For (a+b)(c+d),
≍ (1+1)÷(2+2)
≍ 2÷4 = 1/2
For a/c+a/d+b/c+b/d,
≍ 1/2 + 1/2 + 1/2 + 1/2 = 2 So no it's not true, but invert 2 and it's 1/2 are we close,
can we say:
(a+b)(c+d) = _________1_________
a/c+a/d+b/c+b/d

Where a=1, b=4, c=2, d=8
Testing (a+b)(c+d) = (1+4)(2+8)
=(5)(10) =50
Testing _________1_________ = _________1_________
a/c+a/d+b/c+b/d 1/2+1/8+4/2+4/8
= _________1__________
4/8+1/8+16/8+4/8

= ___1___
25/8
=8/25 ≠ 50 So no that's not it!

R = (360 over W) - 10 - (R.(T -75W) over W)

I need to express the equation isolating R on one side of equation only.

2. see below

3. you have given counter examples
so it's not true

$\displaystyle R=\frac{360}{W}-10-\frac{R(T-75W)}{W}$

then
$\displaystyle R[\frac{T-75W}{W}+1]=\frac{360}{W}-10$

$\displaystyle R[\frac{T-74W}{W}]=\frac{360-10W}{W}$

$\displaystyle R=\frac{360-10W}{T-74W}$

4. ## Cheers

This:$\displaystyle R[\frac{T-75W}{W}+1]$ is the bit I didn't get through… Haven't studied/used much maths for 2 decades, cheers!

Any ideas about a general way to expand $\displaystyle \frac{a+b}{c+d}$ out of interest? Or is that a pointless question, does one just multiple both sides of an equation by (c+d)?

5. add $\displaystyle R\frac{T-75W}{W}$ to both sides, then take R out

can you elaborate the idea of $\displaystyle \frac{a+b}{c+d}$. Because i cannot find anything that is related to it in this equation ?

6. No, no I got it the first time, I just meant I couldn't make that leap before you reply to my original post. Cheers. I actually found a different way of expressing my problem and don't need to resolve that term now so I'm going to leave it. I posted on Dr Math before here, kinda by mistake thinking it was this forum, and someone there produced a series that equates to that term.

Hello,

(a+b)/(c+d) can certainly be written a/(c+d) + b/(c+d)

If you do a long division of (c+d) into a you get

There is a definite pattern here. I don't know whether or not this

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<http://mathforum.org/dr.math/>

The Math Forum @ Drexel is a research and educational enterprise
of Goodwin College at Drexel University: <http://www.drexel.edu/goodwin/>

7. What expression are tying to solve? You have 3 expressions in your original post.

8. Oh they were different bits of a larger set of equation that has something like 10 variables. Sorry, I was jumping around trying to find ways to express the screen co-ordinates of a visualising bit of software for an (old) exercise program called 5BX. (5BX wikipedia).

If you are interested I can email you the documentation and code but it's in a visual programming language called Quartz Composer which is Mac OS X only (free with Apples Developer Tools that ships with every mac). It's just the running section that was proving tricky because the number of steps get's divided by 75 after which 10 star jumps are performed and usually there is and odd number of steps left over after a series of running and jumping blocks. Calculating the time spent on each phase involved simultaeous equations in time equations and transposing those to dimensional values. QC is quite handy at expressing all the interrelated equations as it's a graph of patches that gets evaluated with lazy evaluation each frame (functional language).

Movie of output (10Mb h264):
http://dl.dropbox.com/u/1585739/5BX%20math.mov