>> Welcome to the Cypress College Math Review on Adding
and Subtracting Fractions.
Objective 1: Adding or Subtracting Fractions
with Common Denominators.
Why do we need a common denominator?
Let's take a look at this.
Let's just add two fractions.
Let's add 1/4 and 1/2.
Let's see.
We're adding, so 1 plus 1 is 2, 4 plus 2 is 6, right?
So we get 2 over 6.
No, that's not how you do that.
How do you do it?
OK. So this pizza is broken into four pieces.
So this represents 1/2 of the pizza.
So this person ate 1/4 of the pizza.
Over here, this person was very hungry.
They ate half of a pizza.
Whoa, hope it was a small pizza.
OK. So they ate 1/2.
This represents 1/2 of the pizza.
So we want to add 1/4 and 1/2.
It's definitely not 2/6.
Well, to be able to add these, we need to take this pizza
over here and break it into smaller pieces
in the same size piece that this pizza over here was broken into.
So if we break the 1/2 of the pizza down into quarters,
now we have the same size pieces or the same size parts.
Now we're adding 1/4 and 2/4 and we get 3/4.
So 1/4 plus 2/4 is 3/4.
1/4 plus 1/2 then is also 3/4.
To be able to add or subtract fractions,
they have to have the same denominators.
They have to have the same size parts.
So to be able to add or subtract fractions, you first need
to make sure that each fraction has the same denominator,
so get a common denominator.
Now, let's add some fractions
that already have the same denominator.
For our first example, we have 3/8 plus 2/8.
So we have the same size parts.
They're both 8ths.
So we simply add the numerators.
3/8 plus 2/8 is 5/8.
If this fraction could be reduced, we would reduce it,
but 5/8 does not reduce.
Our next example, 7/9 minus 5/9.
Once again, we have the same denominator,
the same size parts.
So we simply add or subtract the numerators.
So we have 7 minus 5.
7 minus 5 is 2, so our answer is 2/9.
For our next example, we want to add 4/15 and 8/15.
So we have the same size parts, the same denominator.
We can immediately add, 4 plus 8 is 12.
12/15 though-- wait a minute, that can be reduced.
12 and 15 have a common factor.
12 and 15 both have 3 as a factor.
Anytime you can reduce a fraction you need to.
So since 12 and 15 can both be divided
by 3, we need to do that.
12 divided by 3 is 4.
15 divided by 3 is 5.
4 over 5 is our answer.
For the next problem, we have 7/18 minus 17/18.
We have the same size parts, the same denominator,
so we need to subtract.
So we have 7 minus 17.
Well, 7 is smaller, so we're going to end
up with a negative answer.
And we've-- If we subtract 7 from 17, we get 10.
So 7 minus 17 is negative 10.
So we get negative 10 over 18.
Those numbers are both even.
10 and 18 are both even, which means they're divisible by 2.
So we need to simplify this fraction.
2 goes into 10, 5 times.
2 goes into 18, 9 times.
Negative 5 over 9.
5 and 9 do not have any other common factors,
so our answer is negative 5/9.
It doesn't reduce any more.
Pause the video and try these problems.
Objective 2: Finding the Least Common Denominator.
Sometimes the denominators are not the same.
For us to add or subtract fractions
with unlike denominators, we will need to work
with each fraction individually to force the two fractions
to have the same size parts or same denominators.
We will then need to find the least common multiple
of the two denominators.
This is called the LCM, least common multiple.
Since the least common multiple is located in the denominator
of the fraction, then they will change
to the least common denominator.
We call this the least common denominator.
There's many ways to find the least common denominator.
Let's discuss a few of them.
In our first method, we're going to list the multiples
of each denominator until we have one that's the same.
So we're going to list out the multiples of each denominator.
You're looking for the smallest number
that is a multiple of each denominator.
First example, let's find the least common denominator
of this pair of fractions.
3/4 and 5/9.
So we don't look at the numerous,
we're only looking at the denominator.
So we're looking at the numbers 4 and 9.
We want to list out the multiples of 4 and list
out the multiples of 9.
Hint: List out the multiples of the larger number first,
it will make this process faster.
OK, let's list out the multiples of 9.
9 times 1 is 9, 2/9 is 18, 3/9 is 27,
4 times 9 is 36, 5 times 9 is 45.
So these are multiples of the number 9.
Now let's list out multiples of the number 4.
1, 4 is 4, 2, 4 is 8, 3, 4 is 12, 4 times 4 is 16.
We go until we have a number that's the same
as a number on our previous list.
5 times 4 is 20, 6 times 4 is 24, 7 times 4 is 28,
8 times 4 is 32, 9 times 4 is 36.
Bingo, we found one.
So 36 is the smallest common multiple.
It is the least, smallest, common, the same, multiple,
least common multiple.
So the least common denominator of these two fractions is 36.
Now let's find the least common denominator
between 1/8 and 1/18.
So we look at the numbers 8 and 18.
18 is the larger number.
So let's list out the multiples of 18 first.
18 times 1 is 18, 2 times 18 is 36, 3 times 18 is 54,
4 times 18 is 72, 5 times 18 is 90, 6 times 18 is 108.
Hopefully that's enough numbers.
Now let's list out the multiples of 8.
1 times 8 is 8, 2 times 8 is 16, 3 times 8 is 24,
4 times 8 is 32, 5 times 8 is 40, 6 times 8 is 48,
7 times 8 is 56, 8 times 8 is 64, and 9 times 8 is 72.
There we go.
72 is our least common multiple.
We needed to find a number that's a multiple of each
of the numbers that's common to both lists.
So 72 is our answer.
The other method that I'm going
to describe involves finding the prime factors
of each denominator and calculating the LCD
from that factors list.
Sometimes the first method can be quite cumbersome
if the two denominators are large values.
So instead of trying to figure out all the multiples
of each number, we can list the prime factors of each number
and create a combined list of prime factors
that will be the LCD of the fractions.
Let's show this by an example.
Find the least common denominator
for the pair of fractions.
1/8 and 1/18.
We just did this problem so we know the answer ahead of time.
Answer is going to be 72.
All right.
So how do we do this?
We're going to find a prime factorization.
We're going to write out the prime factorization
of each of these numbers.
In other words, break them down into their prime factors.
So 8, we do a little factor tree.
8 is 4 times 2 and 4 is 2 times 2.
These are all prime numbers.
So 8, written out as a product of primes is 2 times 2 times 2.
Now for 18, 18 breaks down into 3 times 6, and 6 is 2 times 3.
These are all prime numbers.
So now we have 18 broken down into a product of primes.
So 18 is 2 times 3 times 3.
Now, we're going to merge these lists of prime factors.
If they share a factor, it's only listed once.
So we notice that this 2 is a common factor.
So we only list that once, and then we have another 2
and another 2, and a 3 and a 3.
If we multiply 2 times 2 times 2 times 3 times 3, we get 72,
which is our least common multiple
or least common denominator.
There's another way to look at this involving exponents.
Let's rewrite the prime factorization.
8 is 2 times 2 times 2,
well that can be written as 2 to the 3rd.
18 is 2 times 3 times 3, so that could be rewritten
as 2 times 3 squared.
So we've written out what's called the prime factorization
of each denominator.
The least common denominator is the product of the largest power
of every factor, the largest power of every factor.
I see a 2.
In fact, I see two 2s.
It doesn't matter if you see 1 or 2 or 3, whatever.
What is the highest power of 2 that we see here?
Well, here we have 2 to the 3rd and we have--
here that we have 2 the 1st.
So 2 to the 3rd is the highest power of 2.
So the least common denominator is going to include 2
to the 3rd, the highest power.
Here I see a 3.
The exponent is a 2.
Over here, I see no 3s, so definitely the highest power
of 3 is 2 that we see there.
So 3 squared is also a factor of a least common denominator.
2 to the 3rd is 8, 3 to the 2nd is 9, 8 times 9 is 72.
This gives us the same answer that we had before.
The LCD is 72.
OK. We now wish to find the least common denominator
between 3/16 and 1/36.
So we're looking at the number 16 and 36.
Let's write out each of these as a product of prime numbers.
Well 16 is 4 times 4, and 4 is 2 times 2.
And since 2 is prime, we're done.
We have 16 written as a product of prime numbers,
2 times 2 times 2 times 2.
36 now. 36 could be written as 6 times 6, and 6 is 3 times 2
and 3 and 2 are prime.
So 36 could be written as 2 times 2 times 3 times 3.
We've written out the prime factorization
of each of these numbers.
So now we wish to merge the two lists.
And if they share a factor, it's only listed once.
Well, here we have a 2, it's listed in both cases,
so we only list at once.
This 2 is also listed on both lists,
so we only include it once.
We have another 2 here, another 2 there, a 3 here and a 3 here.
So the least common multiple
or least common denominator is 2 times 2 times 2 times 2,
times 3 times 3, which gives us 144.
So the least common denominator of 3/16 and 1/36 is 144.
Let's look at this in terms of exponents.
So 16 is 2 times 2 times 2 times 2 or 2 to the 4,
and 36 is 2 times 2 times 3 times 3
or 2 squared times 3 squared.
We want the product of the largest power of every factor.
So I see 2 to the 4 and I see 2 to the 2nd.
So the least common denominator is going to include the 2
to the 4th because that's the larger power.
I see 3 squared and no 3s over here.
So 3 squared is part of our least common denominator.
2 to the 4th is 16, 3 to the 2nd is 9, and 16 times 9 is 144.
Once again, we get 144 as our least common denominator.
Now let's find the least common denominator
between 7/15 and 5/18.
So we're looking at 15 and 18.
15 is 3 times 5.
That's the prime factorization of 15.
18 is 3 times 6, but 6 is 2 times 3.
So 18 can be written as 2 times 3 times 3.
We have it written as a product of primes.
Now we merge the two lists.
We have a 2, we have a 3 that's common, so we just list it once.
We have another 3 and we have a 5.
So the least common multiple
of these two numbers is 2 times 3 times 3 times 5 or 90.
In terms of exponents, once again, 15 is just 3 times 5,
just to the first power for each of them.
And 18 can be written as, 2 times 3 squared.
Now, we want the product of the highest power of every factor.
I see a 2 to the first I see no 2, so we include the 2.
I see 3 to the 2nd and 3 to the 1st,
so we take the higher power 3 to the 2nd.
I see a 5 to the 1st and no 5, so we take the 5.
So the least common denominator is 2 times 3 squared times 5.
3 squared is 9, and 2 times 9 times 5 is 90.
Once again, we get the same answer.
The least common denominator between 7/15 and 5/18 is 90.
Pause the video and try these problems.
Objective 3: Adding and Subtracting fractions
with unlike denominators.
We now know how to find the least common denominator.
Let's use the least common denominator to add
or subtract fractions.
In our first example, we have 3/4 plus 5/24.
Looking at the denominators, we have 4 and 24.
Now, wait a minute, 4 times 6 is 24, so 24 is a multiple of 4.
Since 24 is a multiple of 4, the least common denominator is 24.
Anytime that one number is a multiple of the other number,
the larger number is the least common denominator.
So now we wish to write both of these fractions
with 24 as the denominator.
So the one we have to change is the 3/4.
What would you multiply 4 by to get 24?
In other words, divide 24 by 4, and you get 6.
Yes, 6 times 4 is 24.
Now you can't just multiply the denominator by 6,
you have to also multiply the numerator by 6,
because you can only multiply an expression by the number 1,
something that's equivalent to the number 1.
And in this case, that would be 6 over 6.
We need to multiply the denominator by 6,
which means we have to multiply the numerator by 6.
And we do that we get 18-- 3 times 6 is 18 over 24.
Bingo, now we have the same denominator,
18 over 24 plus 5 over 24.
We add the numerators, 18 plus 5 is 23.
23 over 24 does not reduce,
there's no like factors, so that's our answer.
We now wish to subtract the following fractions,
2/9 minus 5/12.
Looking at the denominators,
let's find the least common denominator between 12 and 9.
Let's write out the multiples of 12 and the multiples of 9.
Let's use this method.
So the multiples of 12 are 1 times 12 is 12,
2 times 12 is 24, 3 times 12 is 36.
I recognize 36 as a multiple of 9,
so I know that's going to be your answer.
The multiples of 9 are 9, 18, 27, 36.
Yes, so the least common multiple
or least common denominator for these fractions is 36.
Now, we're going to write both of the fractions in terms
of 36 as the denominator.
Looking at the 9, what would you need to multiply 9 by to get 36?
In other words, 36 divided by 9 is 4.
So I need to multiply the fraction 2/9 by 4 over 4.
Once again, you can only multiply an expression
by the number 1.
So I'm going to multiply this fraction here by 4 over 4.
Looking at the other fraction 12,
what would you multiply 12 by to get 36?
3. So if we multiply the denominator by 3, we also need
to multiply the numerator by 3.
We're going to multiply 5/12 by 3 over 3.
So now we have 8 over 36 minus 15 over 36.
We subtract these fractions.
The 8 is smallest so we know our answer is going to be negative
and 15 minus 8 is 7,
so our answer will be negative 7 over 36.
7 and 36 do not have any common factors, so we can't reduce it.
So the answer is negative 7 over 36.
For this example, we have 3/5 minus 3/20, we wish to subtract.
5 and 20 are the denominators.
And we notice that 5 times 4 is 20.
So 20 is a multiple of 5.
Once again, anytime one of the denominators is a multiple
of the other denominator,
then that number is the least common denominator.
So 20 is the least common denominator.
We need to write both fractions with 20 as the denominator.
So we look at the 3/5.
What do we need to multiply 5 by?
4. If we multiply the denominator by 4, we also have
to multiply the numerator by 4.
So we multiply by this 4 over 4, which is equal to 1.
3 times 4 is 12, 5 times 4 is 20,
so now we have 12/20 minus 3/20.
We have the same denominator.
So now we subtract the numerators, 12 minus 3 is 9.
9 and 20 do not have any common factors,
so this fraction is reduced.
Our answer is 9 over 20.
For this problem, we have negative 5/12 plus 2/21.
We wish to come up with the least common denominator
between 12 and 21.
Well, 21 is 3 times 7.
12 is 4 times 3, which means that would be 2 squared times 3.
So now we have the prime factorization of 21 and 12.
Let's find the least common denominator based
on these prime factorizations.
Let's see, 2 squared and no 2, so we take the 2 squared,
always the highest power.
We have 3 to the first--
3 to the first, always the highest power.
We take the 3 to the first, they're the same.
We take the 7 also.
So the least common denominator is 2 squared times 3 times 7.
That gives us 4 times 21, which is 84.
The least common denominator is 84.
Now, we have to look at our two fractions and we have
to write them both with the denominator of 84.
You could do it separately and figure out OK,
84 divided by 12 is what?
Or here's a trick.
Our least common denominator is 2 squared times 3 times 7.
12 is 2 squared times 3.
Compare these two products.
The 7 is missing, isn't it?
Ah, so you would multiply 12 by 7
to get the least common denominator of 84.
OK. We multiply the denominator by 7.
We also have to multiply the numerator by 7.
Now let's look at the 21, the other denominator.
21 is 3 times 7.
Compare 3 times 7 with our least common denominator,
2 squared times 3 times 7.
What's missing?
The 2 squared.
Ah yeah, 2 squared, well that's 4.
So we would multiply 21 by 4 to get 84.
We multiply the denominator by 4, which means we have
to multiply the numerator by 4.
5 times 7 is 35.
2 times 4 is 8.
So now we have negative 35 over 84 plus positive 8 over 84.
We have to add these fractions,
we have negative 35/84 plus 8 over 84.
We have a positive and a negative number,
which means we're going to subtract
and take the more dominant side, the answer will be negative
because of the negative 35.
So 35 minus 8 is 27.
So we get negative 27 over 84.
Will that reduce?
Yes it will.
27 is 3 times 3 times 3.
It's made up of 3s.
And 84 is divisible by 3.
You can look at it right here.
You can see that 3 is a factor of 84.
So we divide 27 by 3 to get 9.
We divide 84 by 3 and get 28.
So our answer is negative 9 over 28.
Now, we have 5/6 minus 3/8.
6 is 2 times 3, 8 is 4 times 2, or 2 times 2 times 2 or 2 cubed.
So we have 2 times 3 and 2 to the 3rd.
We look at that.
We want the least common denominator.
2 to the 1st or 2 to the 3rd,
we take 2 to the 3rd, the higher power.
3 to the 1st and no 3s, we take the 3.
So the LCD is 2 cubed times 3, or 8 times 3 which gives us 24.
We need to write both of these fractions
with a denominator of 24.
What would you multiply 6 by to get 24?
Once again, what's missing?
Well, 6 is 2 to the 1st times 3,
and our LCD 24 is 2 to the 3rd times 3.
So we're missing two 2s, yeah, 2 squared or 4.
So we would multiply 6 times 4 to get 24.
If we multiply the denominator by 4,
you have to multiply the numerator by 4.
Now let's look at the 8.
8 is 2 cubed.
Compare 2 cubed with 2 cubed times 3, what's missing?
The 3. So you would multiply 8 by 3 to get 24.
If you multiply the denominator by 3,
you have to multiply the numerator by 3.
5 times 4 is 20, 3 times 3 is 9.
So now we have 20 over 24 minus 9 over 24.
We have the same denominator.
We subtract numerators.
20 minus 9 is 11.
11 and 24 do not have any common factors,
so our answer is 11 over 24.
Pause the video and try these problems.