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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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