Welcome to the presentation on dividing fractions.
Let's get started.
So before I give you the intuition-- actually, I might
do that in a different module-- I'm just going to show you the
mechanics of how you divide a fraction.
And it turns out that it's actually not much
more difficult than multiplying fractions.
If I were to ask you, 1/2 divided by 1/2, whenever you
divide by a fraction, or actually, when you divide by
any number, it's the same thing as multiplying by its inverse.
So 1/2 divided by 1/2 is equal to 1/2 times 2/1.
We just inverted-- inverse-- the second 1/2.
And we know from the multiplication module, 1/2
times 2/1, well, that's just equal to 2/2,
or it's equal to 1.
And that makes sense because, actually, any number divided
by itself is equal to 1.
1/2 divided by 1/2 is 1, just like 5 divided by 5 is 1, just
like 100 divided by 100 is 1.
And this isn't a new principal.
Actually, you were always doing it.
Think about it this way: What is 2 divided by 2?
Well, you know that's 1.
But isn't this also the same thing as 2 times the
inverse of 2, which is 1?
I'll show it to you.
Actually, let me give you a couple more examples to show
that dividing fractions really isn't a new concept, this whole
notion of multiplying by the inverse.
If I were to tell you what is 12 divided by 4?
Well, we know the answer to this, but I'm going to show
you that this is the same thing as 12 times 1/4.
12/1 times 1/4 4 is 12/4, which is 3.
And 12/4 is really just another way of writing 12 divided by 4,
so it's kind of a long way of getting to the same point.
But I just wanted to show you that what we're doing in this
module is nothing new than what we've always been doing
when we divide by a number.
Division is the same thing.
Dividing by a number is the same thing as multiplying by
the inverse of that number.
And just as a review, an inverse, if I have a number
A, the inverse-- inv, short for inverse-- is 1 over A.
So the inverse of 2/3 is 3/2, or the inverse of 5, because 5
is the same thing as 5/1, so the inverse is 1/5.
We're just flipping it.
We're switching the numerator and denominator.
So let's do some fraction division problems.
What is 2/3 divided by 5/6?
Well, we know that this is the same thing as 2/3 times 6/5,
and that's equal to 12/15.
We can divide the numerator and denominator by 3, that's 4/5.
What is 7/8 divided by 1/4?
Well, that's the same thing as 7/8 times 4/1.
Remember, I just flipped this 1/4.
Divide by 1/4 is the same thing as multiplying by 4/1.
That's all you've got to do.
And then we could use a little shortcut we learned in the
multiplication module.
8 divided by 4 is 2.
4 divided by 4 is 1.
So that equals 7/2.
Or if you wanted to write that as a mixed number, this is, of
course, an improper fraction.
Improper fractions have a numerator larger
than the denominator.
If you wanted to write that as a mixed number, 2 goes into 7
three times with a remainder of 1, so that's 3 and a half.
You can write it either way.
I tend to keep it this way because it's
easier to deal with.
Let's do a ton of more problems, or at least as many
more as we can do in the next four or five minutes.
What's negative 2/3 divided by 5/2?
Once again, that's the same thing as minus 2/3-- whoops--
as minus 2/3 times what?
It's times the inverse of 5/2, which is 2/5, and
that equals minus 4/15.
What is 3/2 divided by 1/6?
Well, that's just the same thing as 3/2 times 6/1,
which equals 3 and 1.
We just divided the 6 by 2 and the 2 by 2, so that equals 9.
I think you might be getting it now.
Let's see, let's do a couple more.
And, of course, you can always pause, and look at this whole
presentation again, so you can get confused all over again.
Let's see, let's do minus 5/7 divided by 10/3.
Well, this is the same thing as minus 5/7 times 3/10.
I just multiplied by the inverse.
That's all I keep doing over and over again.
Minus 5 times 3.
Minus 15.
7 times 10 is 70.
If we divide the numerator and the denominator by
5, we get minus 3/14.
We could have also just done it here.
We could have done 5, 2, and we would have gotten
minus 3/14 as well.
Let's do one or two more problems.
I think you kind of get it, though.
Let's say 1/2 divided by minus 3.
Ah-ha!
So what happens when you take a fraction and you divide it by
a whole number or an integer?
Well, we know any whole number can be written as a fraction.
This is the same thing as 1/2 divided by minus 3/1.
And dividing by a fraction is the same thing as multiplying
by it's inverse.
So the inverse of negative 3/1 is negative 1/3, and this
equals negative 1/6.
Let's do it the other way.
What if I had negative 3 divided by 1/2?
Same thing.
Negative 3 is the same thing as minus 3/1 divided by 1/2, which
is the same thing as minus 3/1 times 2/1, which is equal to
minus 6/1, which equals minus 6.
Now, let me give you a little bit of intuition
of why this works.
Let's say I said 2 divided by 1/3.
Well, we know that this is equal to 2/1 times
3/1, which equals 6.
So how does 2, 1/3, and 6 relate?
Well, let's look at it this way.
If I had two pieces of pizza.
I have two pieces of pizza.
Here's my two pieces of pizza right.
Two right here.
So I have two pieces of pizza, and I'm going to divide
them into thirds.
So I'm going to divide each pizza into a third.
I'll draw the little Mercedes sign.
So I'm dividing each pizza into a third, right?
How many pieces do I have?
Let's see, 1, 2, 3, 4, 5, 6.
I have 6 pieces.
So you might want to sit and ponder that for a little bit,
but I think it might make a little bit of sense to you.
Let's do one more just to tire your brain.
If I had negative 7/2 divided by 4/9-- let's pick a negative
4/9-- well, that's the same thing as minus 7/2 times
minus 9/4, right?
I just multiplied by the inverse of negative 4/9.
9 times 7 is equal to-- negative 7 times negative
9 is positive 63, and 2 times 4 is 8.
Hopefully, I think you have a good idea of how to divide by
a fraction now, and you can try out the dividing
fractions modules.
Have fun!