Multiplication of Rational Numbers

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Multiplication of Rational Numbers

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I'm going to warm-up with a couple of straight-up
multiplying fractions or multiplying rational number
problems-- rational number is just another way to say
fractions, it's just any number that can be expressed
as a fraction.
And then we'll actually do a word problem that deals with
the multiplication of fractions.
Let's do part a.
5 over 12 times 9 over 10.
And the fun thing about multiplying and dividing
fractions, at least in my mind, it's easier than adding
or subtracting, you literally just multiply the numerators
when you're multiplying and multiply the denominators.
So you could say that this is the same thing as 5 times 9
over 12 times 10.
And we could just multiply these out, 45 over 120.
But that won't be in simplest form.
We'd be multiplying these big numbers.
And before we do that we can divide the numerator and
denominator by numbers that will simplify it a little bit.
So let's do that.
So if we divide the numerator and the denominator by 5, then
this will become a 1, this will become a 2.
If we divide the numerator and the denominator by 3, this
will become a 3, and this will become a 4.
So the whole thing is simplified to 1 times 3, which
is 3 over 4 times 2, 3/8, which is a lot easier
than 45 over 120.
All right.
Let's just do part c here.
3/4 times 1/3.
We could do that exact same principle.
This is going to be 3 times 1 over 4 times 3.
We can skip a step if we like and cancel out the 3's in the
numerator and the denominator.
So you divide the numerator and denominator by 3.
You'd say that that's a 1, and that's a 1.
And you'd say OK, 1/4 times 1, 1 over 1, is equal to 1/4.
And all I did there, I skipped a step, but all I did, I said,
OK, that's going to be the same thing as 3 times 1
over 4 times 3.
Divide the numerator and the denominator by
3 and you get 1/4.
These are the exact same thing.
I'll do e and f just to finish up things.
So e is 1/13 times 1 over 11.
Well, we just multiply the numerators.
We get 1 over whatever 13 times 11 is.
Let's see, 13 times 10 is 130, and so we're going to have to
get one more 13 in there.
So it's going to be 143, if I did that correctly.
Yeah, 13 times 10 is 130.
Yup.
Looks good to me.
All right, one more, get us really warmed-up.
All right.
I'll do part f.
7 over 27 times 9 over 14.
We can do the same trick we did over here.
Divide the numerator and the denominator by 7, so this
becomes a 1, this becomes a 2.
Then we can divide the numerator and denominator by
9, this becomes a 1, this becomes a 3.
So this becomes 1 times 1 over 3 times 2 or 1/6.
Or you could have said just 7 times 9, which is 63 over 27
times 14, which is a huge number that I don't know the
exact number to and it would have taken you forever to
actually put it in a most simple form.
But this way when you divide it ahead of time it simplified
things a good bit.
So let's do a word problem.
And it involves monkeys.
Three monkeys spend a day gathering
this coconuts together.
When they have finished they're very
tired and fall asleep.
The following morning the first monkey wakes up, not
wishing to disturb his friends, he decides to divide
the coconuts into three equal piles.
There is one left over, so he throws this odd one away,
helps himself to his share and goes home.
So this is the third monkey.
So he divides what he sees into three piles.
We don't know how many are in each pile.
Pile 1, pile 2, pile 3.
And then he had one coconut left over that he chucks.
That's a coconut right there.
That's with that first, he divides the coconuts into
three equal piles.
There was one left over so he throws that away, helps
himself to his share and goes home.
So his share is, let's say, he takes this share and
he throws this away.
That's what this tells us.
He decides to divide the coconuts into three equal
piles, there is one left over, so he throws it away and he
helps himself to his share.
All right.
Now a few minutes later the second monkey awakes, not
realizing that the first has already gone, he, too, divides
the coconuts into three equal heaps.
He finds one left over, throws the odd away, and helps
himself to his fair share.
So this is he finds one left over, throws the odd away,
helps himself to his fair share.
So this is the second monkey.
So this is all he sees.
He just sees these two piles together.
So he divides these into three equal heaps.
So maybe that's one heap right there, that is-- let me make
it a little bit more equal.
So this is one heap, that is two heaps, and that is--
actually, he actually sees one left over, too.
So let me make that clear.
He finds one left over, so he finds one coconut left over,
and then the rest of this-- remember, this is gone and
this is gone.
So this is all we're dealing with at this point.
One left over and three equal heaps.
So one heap, two heaps, and three heaps.
And then he takes one for himself.
He takes one for himself, and he takes the extra coconut and
he throws it away.
So this is all we have left now.
And then you could guess what happens.
In the morning, the third monkey awakes to find that he
is all alone.
He spots the two discarded coconuts and puts
them with the pile.
Interesting.
OK, so he puts them with the pile, giving him
a total of 12 coconuts.
So he spots that one and that one, puts
them back in the pile.
So that is the first-- or maybe I should say-- yeah,
this is the first monkey or three monkeys-- actually, I
should call this the first monkey.
The first monkey did this, the second monkey, and now we're
dealing with the third monkey.
He spots the two discarded coconuts and puts them with
the pile giving a total of 12.
So this is the third monkey, the third one to wake up.
Sorry if I confused you.
This was the first one to wake up, has one left over with
three equal piles.
One left over, keeps one for himself.
Second monkey, he sees these coconuts, he makes three equal
piles, has one left over.
Then the third monkey wakes up, sees these two coconuts,
and then sees what's in the pile and he has
a total of 12 coconuts.
And so they're saying how many coconuts did the first and a
second monkey take?
All right.
Actually, let me be very clear.
The total is 12 coconuts.
This right here is 12 coconuts after taking
the discarded ones.
So this is 12, this right here must be 10 coconuts.
And so this is 10 coconuts, and if you look at these two
piles that are left over, this must be 5 coconuts, 5
coconuts, and he took 5 coconuts.
So if you go to this point over here, how many coconuts
were there?
Well, there must have been 15 plus this 1 coconut over here.
So the first monkey must have had 16 coconuts
in these two piles.
So this must have been 8 and 8.
And then he took an equal share of 8.
So to start off with we had 8 times 3, 24 coconuts plus 1.
So we started with 25 coconuts.
This guy took 8, then we had 16 coconuts.
He took 8, got rid of 1, so he only had 16 left.
The second guy comes along and he said, OK, I'm going to
divide it into 3 groups of 5.
I have one left over.
He took 5.
Then you had 10 left over plus the 2 discarded ones was 12.

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