>> Here we want to add the fractions and make sure that our sum is in the simplest form.
The most important thing to remember
about adding fractions is we have to obtain a common denominator.
So looking at our first example we have 3/5 plus 1/6 so looking at the denominator of 5
and the denominator of 6 the least common denominator is going
to be the least common multiple of 5 and 6 which in this case would be 30.
So we need to rewrite each of these fractions with a denominator of 30.
So I have to multiply the numeric denominator of 3/5 by 6
and multiply the numeric denominator of 1/6 by 5.
Notice how in both cases we're multiplying by a fraction that has a value
of 1 therefore we're just changing the form of the fraction
but they will be equivalent fractions.
So when we multiply these two fractions we're going to have 18/30
and we multiply here we're going to have 5/30.
Now their denominator is the same we can add the fractions.
The denominator is going to stay the same
and then we add the numerators so 18 plus 5 is equal to 23.
In our second example we have 12/27 plus 2/9.
So the first step is to determine our least common denominator which is going
to be the least common multiple of 9 and 27.
So this question is a little bit more challenging.
So what we can do is use the prime factorization of the denominators
to help us determine the least common denominator.
So what we're going to do is rewrite this first fraction as 12 over the prime factorization
of 27 which would be 3 times 3 times 3 plus 2
over the prime factorization of 9 which is 3 times 3.
So now keeping in mind that in order
to add these fractions the denominators must be the same notice how
if this fraction here had an extra factor of 3 the denominators would be the same.
So we can multiply by a factor of 3 here as long as we do the same to the numerator.
So by using the prime factorization of the denominators we can identify the LCD
by determining which factors we must multiply by to make the denominators the same.
Now let's go and rewrite these with our common denominator.
This is going to be 12/27 the original fraction plus this fraction here is now going
to be 6/27 denominator stays the same.
I add the numerator, 12 plus 6 is equal to 18.
Now we can't forget to simplify here.
We saw above the prime factorization of 27 would be 3 times 3 times 3 and the prime factorization
of 18 would be 2 times 3 times 3.
So this simplifies 3 over 3 simplifies to 1 here and here so the final sum here is 2/3.
We'll take a look at one more example in the next video.
In order to subtract fractions we have to have a common denominator
and hopefully we can find the least common denominator,
looking at our first example we have a denominator of 2 and a denominator of 10.
We want to determine the smallest number that is divisible by both 2 and 10.
So in this example the LCD would be 10.
So let's go ahead and rewrite this and leave some space.
So we take 1/2 and multiply the denominator by 5.
We can do that as long as we multiply the numerator by 5 as well.
It's just like multiplying by the fraction of 5 over 5 which will give us 5/10 minus 3/10
and now that the denominator is the same we can go ahead and subtract.
The denominator stays the same and we subtract the numerator so 5 minus 3 is equal to 2.
So the difference is 2/10 but 2/10 does simplify.
We can rewrite 10 as 2 times 5.
Notice how they have a common factor of 2 so our simplified difference is 1/5.
Remember these are not disappearing they're simplifying to 1s.
Now let's look at a second example.
This one is going to be a little bit more challenging
to determine the least common denominator and when we have a challenging problem
like this it's helpful to determine the least common denominator by looking
at the prime factorization of the denominators.
So I'm going to go ahead and rewrite these two fractions
with the denominators in prime factored form.
So we're going to have 19 all over the prime factorization of 42,
42 is 6 times 7 so we'll have 2 times 3 times 7 minus 13 over the prime factorization of 70.
Well 70 would be 10 times 7 and 10 is 2 times 5 so of 2 times 5 times 7.
Now we can use these prime factors to help build the least common denominator meaning if we want
to subtract these fractions the denominators must be the same
and therefore contain the same factors.
So starting with this first denominator we can ask what factors does this denominator have
that this one doesn't have?
Notice how this one has a factor of 5 but this one doesn't.
So it's going to have to have a factor of 5 if the denominators are going to be the same.
So multiply this by 5.
We can do this as long as we do the same to the numerator.
Now we can go to the second fraction and ask, what does this first denominator have
that the second denominator doesn't have?
Notice how this has a factor of 3 and this one doesn't so it must contain a factor of 3
if the denominators are going to be the same.
So multiply the numerator by 3 as well.
Notice how now we have a common denominator
because both the nominators contain the same prime factors.
So let's go ahead and rewrite this, 19 times 5 is going to be 95.
So we have 95 all over 2 times 3 times 7 times 5 which is equal to 210 minus,
here we're going to have 13 times 3.
That's 39 all over the common denominator of 210.
Now that we have a common denominator we can subtract.
The denominator stays the same.
The numerator will be 95 minus 39.
That's 56 and of course the last step is to simplify this fraction.
Let's go ahead and write off the prime factorization of the numerator denominator,
so let's go ahead and clear this out.
Here's the prime factorization of 56.
You have 3 factors of 2 and a factor of 7 and for 210 we'll have 21 times 10 to start
so a factor of 2, a factor of 3, a factor of 5 and a factor of 7.
Now we can simplify this fraction and now we can simplify 2
over 2 simplifies to 1, 7 over 7 simplifies to 1.
So the simplified difference is 4/15. ------------------------------355a30dbabc0--