In elementary school we are taught to add/subtract fractions in a way that, quite frankly, is a BAD WAY of adding/subtracting fractions! If you don't know what I'm talking about, here's a short review...
We have a fraction, say (1/2). Let's add a third. We have (1/2) + (1/3).
So, what's the first step?
Well, in elementary school you were probably taught to cross multiply. Let's try it
We first get (1)(3) = 3 and (1)(2) = 2
Adding these together gets us 3 + 2 = 5
Now we multiply the denominators, getting (2)(3) = 6
We can now put these two numbers together: (5/6)
Though this method works, it's not the best way to go about adding two different fractions.
First off, why the heck does this work?!
Though it seems like magic, there's a method.
First, we cross multiplied. Putting this in a way that shows the whole expression gives
(3)(1) + (2)(1) = 3 + 2
(2) (3) 2 3
We also multiplied the denominators:
(3) + (2) = 3 + 2
(2)(3) (3)(2) 6 6
Now that we have similar denominators, one can see the obvious next step
3 + 2 = 3+2 = 5
6 6 6 6
which brings me to my next method (easier in my opinion, although not for everyone!).
This is very similar to what the previous method does. Actually, it's the exact same thing! Here it goes...
We can use the same fractions we started with.
So, we have (1/3) + (1/2).
Here's step 1: multiply each fraction by its partner's denominator over the same denominator. Here's what I mean...
1 (3) + 1 (2) = 3 + 2 = 5
2 (3) 3 (2) 6 6 6
That wasn't so hard now, was it? No cross multiplying, no adding 'random' numbers together.
All you are doing is finding a common multiple of the denominators. This works with any arithmetic of quotients. More abstractly...
(a/b) + (c/d) = (a(d)/b(d)) + (c(b)/d(b)) = (ad/bd) + (cb/bd) = (ad+cb)/(bd)
And there you have it!
So next time you have a MONSTER fraction with ten variables and high degree polynomials, you may feel a lot better looking at it!