Subtracting Fractions

Date: 08/03/2001 at 13:06:12
From: Albana 
Subject: Subtracting fractions

How do you subtract fractions?

Date: 08/03/2001 at 22:50:00
From: Doctor Jeremiah
Subject: Re: Subtracting fractions

Hi Albana,

Thanks for writing.

Subtracting fractions is the hardest thing you can do with fractions,  
but there is a secret that will help.

You can't add or subtract fractions when the bottom numbers (the 
denominators) are not the same. The secret is to make the denominators 
the same. For example:

2/3 - 5/8

How do we deal with this? We can't subtract because the denominators 
are different. We need to find a number we can use for the denominator 
that will be the same for both fractions; a number that is a multiple 
of both 3 and 8.

We could look for the lowest common multiple but that is too much 
work, so we will use the multiple that is easiest to calculate, which 
is the product of the denominators (3 times 8). 

Okay, now we know what denominator we need to use (24) but how do we
use it?

Well, if we want to make our answer right we can't change the meaning 
of the fraction. That means we can only multiply by one or add a zero, 
but nothing else.

To change the fraction's denominator of 3 into a denominator of 24 we 
must multiply it by whatever value makes the number 3 change to a 24 
(3 times 8 = 24).

But wait! Remember, we can't change the meaning of the fraction. We 
need to do something really sneaky for it to work. We need to multiply 
2/3 by one, but not just the number '1'. If you remember, anything 
divided by itself is one, so 8/8 is one. We can multiply 2/3 by 8/8 
with no problems.

It turns out that 2/3 * 8/8 = (2*8)/(3*8) = 16/24. We have changed the 
2/3 into a fraction with a denominator of 24. Just what we wanted.

Now if we can just change 5/8 into a fraction with a denominator of 24 
we will be almost done. Again, we need to multiply by one. This time 
we need to multiply the denominator by a value that will change 8 into 
24 (8 times 3 = 24 so we need to multiply 5/8 by 3/3).

It turns out that 5/8 * 3/3 = (5*3)/(8*3) = 15/24. We have changed the 
5/8 into a fraction with a denominator of 24. Just what we wanted.

So now we can do the subtraction because 16/24 has the same 
denominator as 15/24.

2/3 - 5/8 = 16/24 - 15/24 = 1/24

Keep in mind that we should normally try to simplify our answer, but 
in this example 1/24 cannot be simplified any further, so the answer 
is 1/24.

Adding fractions is done the same way: you need to find a common 
denominator first.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/