Adding and Subtracting Fractions
lowest common
denominator  adding & subtracting
Lowest Common Denominator
To be able to compare two fractions (or add them together, or subtract
one from the other) they need to have the same denominator.
We know that multiplying by 1 won't change the value of a number, so
if we multiply a fraction by a fraction 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}
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.
Adding & Subtracting Fractions
To be able to add two fractions together they need the same denominator.
If the denominators are not the same we need to make them the same by
the method mentioned above. Let's look at an example.
^{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.
^{4}⁄_{12} + ^{3}⁄_{12} = ^{7}⁄_{12}
As I mentioned earlier, sometimes the denominators will have common
factors. 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}
The ice cream cake example above was actually an example where we subtracted
one fraction from another.
^{2}⁄_{9}  ^{5}⁄_{24} = ^{16}⁄_{72}  ^{15}⁄_{72} = ^{1}⁄_{72}
