Subtracting Fractions
Objective
So what is a fraction?
So before we even begin looking at adding fractions, let’s look at what a fraction actually is.
So when you see a fraction, what mathematical procedure comes to mind?
Division
Whenever you see a fraction think division.
So why did you have such a hard time dealing with fractions?
Because you’ve been doing two procedures at once, that’s why.
So, we will go over fractions, but
Give yourself a break
We’ll get through this.
Adding Fractions
So let’s look at what a fraction is:
A fraction of something is only a portion of that thing.
Or in other words, like said before, a division of a thing.
So how do we add fractions?
Well, we have to change the fractions some.
So, let’s go over that.
First, let’s look at something.
Let’s say we have something like this:
Now, what do we know that this reduces to?
But, the question becomes, did we change anything?
I mean, if we have 3 third slices of pizza, do we still have a full pizza?
So then, we can see that 3/3 is actually 1 in disguise.
Why bring this up?
Because what is the one number we can multiply any number by, and not change anything?
That’s right, 1.
Finding the Least Common Multiple
So here’s how we add fractions.
Let’s start off with an example:
We can’t just add these right?
We have no idea how to add them.
However, what if we changed them to something we can add?
But, we can’t change the equation right?
So, how do we change the numbers, without changing the amount?
Well, we can try to multiply the fractions by 1, but what kind of one?
This is where the Least Common Multiple comes in.
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 2:
2 4 6 8 10 12 14 16 18 20
Now, let’s list out the multiples of 3:
3 6 9 12 15 18 21 24 27 30
Now, which number do they both share?
6
6
Now we have to find out how to make 3 into 6.
We multiply it by 2!
Now we have to find out how to make 2 into 6.
We multiply it by 3!
THE CHEATING WAY
So, there is actually another way of creating a common denominator instead of finding the least common multiple
But, it’s not the best way, however, it will work every single time.
We multiply by opposite denominators.
Sounds weird, but let’s try it:
So, what we do, is look at:
( )
= 35
Now, we multiply each side by the opposite number to get 35:
Again, this works every time, but it can take more work.
THAT’S ALL THERE IS TO IT!
That’s how we add fractions!
We need to find the least common multiple, figure out what to multiply the fractions by, then add them.
This is also how we subtract fractions.
So, let’s look at a few more examples:
EXAMPLE 1:
Let’s say we have:
Again, we can’t just add these right?
We need to find the least common multiple first!
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 5:
5 10 15 20 25 30 35 40 45
Now, let’s list out the multiples of 4:
4 8 12 16 20 24 28 32 36 40
Now, which number do they both share?
20
20
Now we have to find out how to make 5 into 20.
We multiply it by 4!
Now we have to find out how to make 4 into 20.
We multiply it by 5!
HOW DO WE SUBTRACT FRACTIONS?
Well, remember, subtraction is just adding, but adding the other numbers opposite.
And we know how to add fractions now, right?
First, we need to find a least common multiple
Then we need to multiply the fraction by 1, but a different form of 1 so we have the same denominator
Then we subtract them (add the opposite).
See, we’ve already done this before, so let’s look at a few examples:
EXAMPLE 1:
Let’s say we have:
Again, we can’t just subtract these right?
We need to find the least common multiple first!
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 5:
5 10 15 20 25 30 35 40 45
Now, let’s list out the multiples of 4:
4 8 12 16 20 24 28 32 36 40
Now, which number do they both share?
20
20
Now we have to find out how to make 5 into 20.
We multiply it by 4!
Now we have to find out how to make 4 into 20.
We multiply it by 5!
Now, let’s make sure to add it’s opposite:
SO, IT’S BASICALLY WHAT WE’VE BEEN DOING?
Basically, we’re just subtracting the way we did before, just now there’s a denominator to take into consideration.
But, essentially, we’re doing the same old same old.
So, let’s try a few more:
EXAMPLE 2:
Let’s say we have:
Again, we can’t just subtract these right?
We need to find the least common multiple first!
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 3:
3 6 9 12 15 18 21 24 27
Now, let’s list out the multiples of 4:
4 8 12 16 20 24 28 32 36 40
Now, which number do they both share?
12
12
Now we have to find out how to make 3 into 12.
We multiply it by 4!
Now we have to find out how to make 4 into 12.
We multiply it by 3!
Now, let’s make sure to add it’s opposite:
Example 3:
Let’s say we have:
Again, we can’t just subtract these right?
We need to find the least common multiple first!
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 11:
11 22 33 44 55 66 77 88 99
Now, let’s list out the multiples of 5:
5 10 15 20 25 30 35 40 45 50 55
Now, which number do they both share?
55
55
Now we have to find out how to make 11 into 55.
We multiply it by 5!
Now we have to find out how to make 5 into 55.
We multiply it by 11!
Now, let’s make sure to add it’s opposite:
Example 4:
Let’s say we have:
Again, we can’t just subtract these right?
We need to find the least common multiple first!
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 5:
5 10 15 20 25 30 35 40 45
Now, let’s list out the multiples of 4:
4 8 12 16 20 24 28 32 36 40
Now, which number do they both share?
20
20
Now we have to find out how to make 5 into 20.
We multiply it by 4!
Now we have to find out how to make 4 into 20.
We multiply it by 5!
Now, let’s make sure to add it’s opposite: