Dividing fractions Dividing fractions
- 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.
- 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.
- 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,
- 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.
- 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!
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