Math Is Fun Forum

planetdomi

Member

Registered: 2011-09-17

Posts: 16

rearranging fractions

Hi,

Im doing a Maths course and I am finding the basics very difficult. the last time I did Maths was at O Level over 20 years ago and my poor ancient mind seems to have forgotten most of it.

Anyway can anyone help with the following.

Why and how does p+q/pq become 1/q+1/p

How is it possible to simplify (x-√3)(x+√3) to x² - 3

Thanks

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: rearranging fractions

Hi planetdomi,

For your first question, since pq is a product, we can say:

We can split the fraction in this way because we have addition on the top and not on the bottom. If you want to hear more about why, then just ask, from there we can simply cancel the p's and q's:

Since p is common to p and pq and q is common to q and pq.

We can simplify your fraction by expanding the brackets. I was always taught FOIL - first, outside, inside, last, which means that all your like-terms are already collected for you, but you can multiply them however you wish:

This is because x√3 - x√3 obviously gives 0 and √3 times √3 is 3, by the definition of a square root.

Your last question, by the way, is an example of what is sometimes called 'the difference of two squares'. If you have (x - a)(x + a), you will always get:

As you can see by using foil. Since x and a can be any number, we can see that this is always true. This is very useful, if you ever see something of the form:

You can easily factorise it using this rule. So for example:

Is that any clearer?

Last edited by Au101 (2011-09-22 00:14:22)

planetdomi

Member

Registered: 2011-09-17

Posts: 16

Re: rearranging fractions

Fantastic
thanks for all your help
If you have any time left then I would like to hear why you can split the fraction because of addition on top and not the bottom.

thanks again

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: rearranging fractions

Well, let's look at why we can't split the fraction up if we have p + q as our denominator. Since p and q are variables, they can stand for any number, so let's choose a simple example where p is 3 and q is 4. If we have:

Then thats:

Do you see? If we have a product, though, we're okay, because of the rules for addition of fractions. Do you see that above, we can only add to fractions with the same denominator, so i had to use the lowest common denominator of 12, and do:

We can do this for any number, so if we want to do:

Therefore:

It doesn't matter what ad and bc are, we can always split them, if it would help, let's use p and q:

We can split fractions where the addition is on the top, like this, because they have the same denominator. We are just undoing the combination when we add fractions, if you see what i mean?

planetdomi

Member

Registered: 2011-09-17

Posts: 16

Re: rearranging fractions

Wow
Thanks for all your hard work.
Makes perfect sense