>> Hello, and welcome to a video on adding and subtracting fractions.
The goals of this video will be to add and subtract fractions as well
as to verify the addition and subtraction of fractions on the graphing calculator.
Let's go and take a look at the rules for adding and subtracting fractions.
Notice if we have a common denominator and we're adding two fractions,
it's as easy as adding the numerators and keeping the denominator the same.
And for subtraction it's exactly the same, except we would be subtracting our numerators.
Denominator, again, stays the same.
But notice the key here is, in order to add or subtract,
we do have to have a common denominator.
To add or subtract when denominators are different,
we will need to find a common denominator.
And that is, well, we need to find the least common multiple of the denominators.
That number will be the least common denominator or LCD.
Next we will multiply by 1, using the appropriate notation n over n
to express each number in terms of the LCD.
Then we will add or subtract the numerators, keeping the denominator the same.
And then of course, we will simplify, if possible.
I'd like to spend a moment explaining why it's so important
that we do find a common denominator.
Let's say, for example, we consider 1/2 plus 1/4.
Here's a common error that I see: 1/2 plus 1/4, well, maybe we should just add across the top
and add across the bottom, and why can't that be our answer?
One plus 1 would be 2, 2 plus 4 would be 6.
So why can't 1/2 plus 1/4 equal 2/6?
Well, here's a geometric model for why that does not work.
Here's a model for 1/2, here's a model for 1/4, and if I was to take these two and combine them
into a single fraction, it doesn't make sense
that 2/6 would be our answer for a variety of reasons.
First of all, thinking of this in terms of area, this area here is less
than the original fraction of 1/2, so there's no way that we could add 1/2 plus 1/4
and get something smaller than 1/2.
The sum of these two fractions should represent the area
of combining this rectangle with this rectangle.
But in this case, it doesn't.
So this idea is invalid.
Let's take a look at this another way.
Here's a model for 1/2; here's a model for 1/4.
Now, what I've done here is I've taken the fraction for 1/2 and I divided it into fourths.
And now you can see that these two fractions,
each piece is the same size; it represents one fourth.
Two-fourths equaling 1/2, and this fraction is obviously 1/4.
Now, if I was to combine these pieces,
obviously I would have three equal pieces, each piece being a fourth.
So if I add these two together it makes sense I'd have three pieces,
so the answer would have to be 3/4.
So this kind of emphasizes the importance of finding that common denominator.
If we rewrite 1/2 as 2/4, modeled here, and we add in 1/4, if our denominators are the same,
we can just simply add the numerators.
So that's the key: finding a common denominator before we add or subtract.
Let's take a look at some examples.
Three-eighths plus 5/6, we need to find the least common multiple of 8 and 6.
There are a couple of ways of doing this.
If you can't just look at that and identify the LCD as 24,
I'll show you a way that may be a little bit easier.
If we write out the denominators in prime factored form...
...6 would be 2 times 3.
Essentially what we have to do is make sure that the denominators contain the same factors.
And the way we do that is we start with this denominator and say,
"What does this denominator have that this one doesn't?"
And you can see it's missing a factor of 3.
I can multiply this denominator by a 3 as long as I do the same to the numerator.
And there's that term in the notes of multiplying by n
over n. This is just equal to the value of 1.
Now we go to the second fraction and see which factors are missing
in this denominator that this one has.
It already has the 3, it has one 2, but in order for them to be the same,
it's going to need two more factors of 2.
Now, both fractions contain the same factors; therefore, we have found the common denominator,
and in fact, we have found the least common denominator of 24, indirectly.
Let's multiply this back together now.
We have 9/24, 2 times 2 times 2 times 3 gives us the 24 --
plus this will be 5 times 2 times 2 -- that would give us 20 over 24.
Now we simply add the numerators: 20 plus 9 of course is 29.
And that is fine as an answer: 29 is a prime number.
If they do want us to write this as a mixed number, it would be 1-5/24.
Taking a look at the next example, again, finding the LCD of 12 and 18 may be challenging.
If you know your multiplication tables really well,
you can probably see that it is going to be 36.
But if you can't, let's go ahead and try this prime factor method again.
If I break down 12 into its primes, it will be 2 times 2 times 3.
If I break down 18 into its primes, that would be 2 times 3 times 3.
Again, in order to have a common denominator, these have to contain the same factors.
So we can ask ourselves, what factors do we see
in the second denominator that this one does not have?
It already has a 2 but it doesn't have two 3s.
It needs another one.
And I can do that just as long as I multiply the top by 3 as well.
Now I ask the same question for the second fraction.
What factors does this denominator contain that this one doesn't?
And you can see it needs another factor of 2.
Okay, so we're essentially multiplying by 1; they both contain two 2s and two 3s.
That would be 9 times 4; again, as we mentioned, that would be 36.
So we have 21/36 minus 26/36.
Okay, now this difference is a negative value.
Twenty-one minus 26 would be a negative 5 with a denominator of 36.
Let's go ahead and verify this one on the graphing calculator.
I'm going to enter in 7/12 in a set of parentheses, and then I will subtract 13/18.
Again, this will give me the decimal value, but if I hit Math, Enter, Enter,
it will convert it back to a fraction for me.
That verifies all of our hard work.
Now, notice I threw in a couple examples here with mixed numbers.
Now, I will talk about other ways to add and subtract mixed numbers,
but one way is to simply convert them to improper fractions.
So in this case, 3 times 4 plus 2, that'll be 14/3 plus 19/5.
Now, this problem is a little bit easier to find the LCD.
Three times 5 would give us 15.
If you ever get stuck on finding a common denominator, one method you could use is
to just multiply the denominators together.
It may not be the least common denominator, but it will always be a common denominator.
So what that tells us is, if I multiply this fraction by 5 over 5, and this fraction by 3
over 3, I would have a common denominator of 15, okay?
So we have 70/15 plus 57/15.
We have our common denominator.
We can go ahead and add these two fractions together: 70 plus 57 would equal 127/15.
Since the original problem was a mixed number, we should give this back as a mixed number.
So what we would need to do now is take 127, divide it by 15.
Fifteen goes into 127, 8 times, with the remainder 7; therefore,
our mixed number would be 8-7/15.
I hope that helps.
Have a great day. ------------------------------de05dd3aeb7e--