Adding and Subtracting Fractions
adding & subtracting  lowest common
denominator
Adding & Subtracting Fractions
To be able to directly compare two fractions (or add them together, or subtract
one from the other) they need to have the same denominator. When fractions have the same denominators they are basically the same sorts of things, meaning they can be combined. It's like combining two piles of bananas, instead of a pile of pears and a pile of bicycles.
If the denominators are not the same we need to make them the same by
multiplying by a unity fraction. The simplest way is using a unity fraction made from the denominator of the other fraction.
^{1}⁄_{3} + ^{1}⁄_{4} =
( ^{1}⁄_{3} x ^{4}⁄_{4} )
+ ( ^{1}⁄_{4} x ^{3}⁄_{3} )
= ^{4}⁄_{12} + ^{3}⁄_{12}
Once the denominators are the same the numerators can simply be added
together while the denominator stays the same. This example has four "somethings" added to three "somethings" giving seven "somethings". In this case the "somethings" are twelfths, but they could easily be bananas.
^{4}⁄_{12} + ^{3}⁄_{12} = ^{7}⁄_{12}
Sometimes the denominators will have common
factors, and we need to simplify the final fraction. For example:
^{1}⁄_{2} + ^{1}⁄_{4} =
( ^{1}⁄_{2} x ^{4}⁄_{4} )
+ ( ^{1}⁄_{4} x ^{2}⁄_{2} )
= ^{4}⁄_{8} + ^{2}⁄_{8} = ^{6}⁄_{8}
Then simplify ^{ 6}⁄_{8} = ^{3}⁄_{4}
Or we could use a better multiplying fraction to start with, so we can
use the lowest common denominator.
^{1}⁄_{2} + ^{1}⁄_{4} =
( ^{1}⁄_{2} x ^{2}⁄_{2} )
+ ^{1}^{}⁄_{4} = ^{2}⁄_{4} + ^{1}⁄_{4} = ^{3}⁄_{4}
Lowest Common Denominator
We know that multiplying by 1 won't change the value of a number, so
if we multiply a fraction by a fraction which is equal to 1, it won't change in
value. There are lots to chose from, and we can pick any we like.
1 = ^{1}⁄_{1} = ^{2}⁄_{2} = ^{3}⁄_{3} =
... = 
6,782 
= ... 

6,782 
Let's look at an example. Assuming you're hungry, would you rather have ^{2}⁄_{9} of
an ice cream cake or ^{5}⁄_{24} of it? To be sure of
which is bigger we'll have to convert them so they have the same denominator.
The easiest way to choose one is to use the denominator of the other
fraction.
^{2}⁄_{9} x ^{24}⁄_{24} = ^{48}⁄_{216}
^{5}^{}⁄_{24} x ^{9}⁄_{9} = ^{45}⁄_{216}
So ^{2}⁄_{9} > ^{5}⁄_{24} by ^{3}⁄_{216}
But 216 is divisible by 3 (because it's 6^{3} and 6 is divisible
by 3, and because 2+1+6 is a multiple of 3) which means there were common
factors in the denominators. In this case, both 9 and 24 can be divided
by 3. Finding the lowest common multiple of the original two denominators
will give us the lowest common denominator, which will give us
the simplest fraction at the end of it.
^{2}⁄_{9} x ^{8}⁄_{8} = ^{16}⁄_{72}
^{5}^{}⁄_{24} x ^{3}⁄_{3} = ^{15}⁄_{72}
^{2}⁄_{9}  ^{5}⁄_{24} = ^{16}⁄_{72}  ^{15}⁄_{72} = ^{1}⁄_{72}
So ^{2}⁄_{9} > ^{5}⁄_{24} by ^{1}⁄_{72}
^{1}⁄_{72} of an ice cream cake isn't much, but it's
still worth having.
"Lowest common denominator" is also used figuratively
to describe how television sometimes creates their programmes to appeal
to the lowest ideals of their viewing audience, for example by using
the simplest, coarsest humour, instead of anything that might require
careful consideration.
