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Understanding division of fractions

Using a number line, we'll explain why multiplying by the inverse is the same as dividing. Created by Sal Khan.

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  • starky ultimate style avatar for user Chris Williams
    OK, here we go with my question. I easily understand how to multiply and divide fractions. I have watched these videos over and over and still do not understand conceptually WHY I flip the reciprocal and multiply across to get the answer. Is there another source to read or watch to explain why these steps work and what is actually happening?
    (14 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      This is a great question and I'm glad you are thinking at a deeper level in mathematics. The properties of numbers are useful for explaining why we flip the second fraction and multiply, when we divide fractions.

      For b, c, and d nonzero, the problem a/b divided by c/d is the answer to the question "what number times c/d is a/b?" This is because of division's inverse relationship with multiplication (basic definition of division), but now applied to fractions.

      Because any number times 1 is itself, d/c * c/d = dc/cd = cd/cd = 1, and factors can be grouped in any order (associative property of multiplication), we have

      a/b = a/b * 1 = a/b * (d/c * c/d) = (a/b * d/c) * c/d.

      So a/b * d/c is the answer to the question "what number times c/d is a/b?"

      Therefore, a/b divided by c/d equals a/b * d/c.
      (3 votes)
  • aqualine sapling style avatar for user Cody Reed
    Its kinda confusing but im getting the hang of it thanks for the video!
    (5 votes)
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  • starky sapling style avatar for user Rose Whaley
    hi I just watched this and none of this made sense to me could really slow down the process
    (5 votes)
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  • leafers tree style avatar for user Anthony
    how is 6/9 changed into 2/3
    (5 votes)
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  • aqualine ultimate style avatar for user robintan
    I thought the anwer was 8/3 why is it not?
    (3 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      If two fractions with a common denominator are divided, the common denominator is canceled out instead of kept (unlike addition or subtraction of fractions).

      Note that for positive numbers, dividing by a number less than 1 gives a larger answer. Because 1/3 is less than 1, 8/3 divided by 1/3 is greater than 8/3, not equal to 8/3. So the answer 8/3 actually would not make good sense.

      Another way to think of 8/3 divided by 1/3 is to answer the question “how many 1/3’s are in 8/3”? Clearly there are 8 of them, so the answer is 8.
      (4 votes)
  • sneak peak green style avatar for user Abdirahman Sheikh Hassan
    I understand how 8/3 / 1/3 is equal to 8; 8/3 is an improper fraction that is greater than 1. But if we divide 1/3 / 8/3, the answer is 1/8. How can we show that on a number line?
    (3 votes)
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  • male robot hal style avatar for user BlackTigerz
    At :45 seconds in the video Sal said 8/3 divided by 3 but shan't it be 8/3 divided by 1/3
    (3 votes)
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  • blobby blue style avatar for user krystall
    how do you divide fraction without a number line
    (3 votes)
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  • duskpin ultimate style avatar for user Jack Smith
    Can't you also divide 8 and 1 and 3 and 3 to get the first answer, and divide 8 and 2 and 3 and 3 to get the second answer?
    (5 votes)
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    • leaf green style avatar for user S Petty
      Actually, this method will work ONLY if you have common denominators. It's easier to do it the other way. But you can find common denominators, which cancel each other out, then the answer is the first numerator divided by the second numerator. For example,
      2/3 ÷ 5/4 = 8/12 ÷ 15/12 = 8/15 over 12/12 or 8/15 (since 12 divided by 12 is 1)
      You can also do it without common denominators, but then you're left with
      2/5
      ----
      3/4

      which is another dividing fractions problem, so doing it without the common denominator creates an infinite loop of dividing fractions, and no one wants that!
      (0 votes)
  • blobby green style avatar for user Kinaya Aruthur
    The video is kind of hard to understand :/
    (4 votes)
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Video transcript

Let's think about what it means to take 8/3 and divide it by 1/3. So let me draw a number line here. So there is my number line. This is 0. This is 1. And this is 2. Maybe this is 3 right over here. And let me plot 8/3. So to do that, I just need to break up each whole into thirds. So let's see. That's 1/3, 2/3, 3/3, 4/3, 5/3, 6/3, 7/3, 8/3. So right over here. And then of course, 9/3 would get us to 3. So this right over here is 8/3. Now, one way to think about 8/3 divided by 3 is what if we take this length. And we say, how many jumps would it take to get there, if we're doing it in jumps of 1/3? Or essentially, we're breaking this up. If we were to break up 8/3 into sections of 1/3, how many sections would I have, or how many jumps would I have? Well, let's think about that. If we're trying to take jumps of 1/3, we're going to have to go 1, 2, 3, 4, 5, 6, 7, 8 jumps. So we could view this as-- let me do this in a different color. I'll do it in this orange. So we took these 8 jumps right over here. So we could view 8/3 divided by 1/3 as being equal to 8. Now, why does this actually make sense? Well, when you're dividing things into thirds, for every whole, you're now going to have 3 jumps. So whatever value you're trying to get to, you're going to have that number times 3 jumps. So another way of thinking about it is that 8/3 divided by 1/3 is the same thing as 8/3 times 3. And we could either write it like this. We could write times 3 like that. Or, if we want to write 3 as a fraction, we know that 3 is the same thing as 3/1. And we already know how to multiply fractions. Multiply the numerators. 8 times 3. So you have 8-- let me do that that same color. You have 8 times 3 in the numerator now, 8 times 3. And then you have 3 times 1 in the denominator. Which would give you 24/3, which is the same thing as 24 divided by 3, which once again is equal to 8. Now let's see if this still makes sense. Instead of dividing by 1/3, if we were to divide by 2/3. So let's think about what 8/3 divided by 2/3 is. Well, once again, this is like asking the question, if we wanted to break up this section from 0 to 8/3 into sections of 2/3, or jumps of 2/3, how many sections, or how many jumps, would I have to make? Well, think about it. 1 jump-- we'll do this in a different color. We could make 1 jump. No, that's the same color as my 8/3. We could do 1 jump. My computer is doing something strange. We could do 1 jump, 2 jumps, 3 jumps, and 4 jumps. So we see 8/3 divided by 2/3 is equal to 4. Now, does this make sense in this world right over here? Well, if we take 8/3 and we do the same thing, saying hey, look, dividing by a fraction is the same thing as multiplying by a reciprocal. Well, let's multiply by 3/2. Let's multiply by the reciprocal of 2/3. So we swap the numerator and the denominator. So we multiply it times 3/2. And then what do we get? In the numerator, once again, we get 8 times 3, which is 24. And in the denominator, we get 3 times 2, which is 6. So now we get 24 divided by 6 is equal to 4. Now, does it make sense that we got half the answer? If you think about the difference between what we did here and what we did here, these are almost the same, except here we really just didn't divide. Or you could say you divided by 1, while here you divided by 2. Well, does that make sense? Well, sure. Because here you jumped twice as far. So you had to take half the number of steps. And so in the first example, you saw why it makes sense to multiply by 3. When you divide by a fraction, for every whole, you're making 3 jumps. So that's why when you divide by this fraction, or whatever is in the denominator, you multiply by it. And now when the numerator is greater than 1, every jump you're going twice as far as you did in this first one right over here. And so you would have to do half as many jumps. Hopefully that makes sense. It's easy to think about just mechanically how to divide fractions. Taking 8/3 divided by 1/3 is the same thing as 8/3 times 3/1. Or 8/3 divided by 2/3 is the same thing as 8/3 times 3/2. But hopefully this video gives you a little bit more of an intuition of why this is the case.