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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?
    (15 votes)
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    • hopper cool style avatar for user Philip
      You have a very good question. Think about it this way: a fraction itself is a division problem, the numerator divided by the denominator. When you multiply by a value greater than 1, the original amount becomes greater; when you multiply by a value less than 1, the original value becomes smaller.
      You want to find a way to "move the dividend into the denominator of the divisor". The easiest way to do this is to change the dividend into its inverse. For whole numbers, the value moves to the denominator, and the "invisible 1" (since anything divided by 1 is itself) goes to the numerator. To find the numerator you multiply the numerators together; to find the denominator you multiply the denominators together.
      The same works for values which are not whole numbers.

      Here are two examples:
      10÷(2/5). The 2 is a whole number, which is already being divided by 5. Division by a number greater than 1 means the value is being "reduced" (and 10÷2 means the 10 is getting reduced to one-half). But at the same time, the "reduction power of the 2" is also being reduced, reduced to one-fifth. So in order to compensate for that, you will need to multiply the 10 by a 5 as well. In the end, there will be a 5 in the numerator and a 2 in denominator, and the quotient of the problem is 25.

      When you divide by a fraction greater than 1, the original value is still reduced. Let's say you have 60÷(5/4). Without the "1/4", the 60 would be divided by 5; but the "1/4" reduces the "reduction power", and the quotient will be 48 in the end.

      Hope this helps in better understanding and clearing things up. [R]
      (15 votes)
  • blobby green style avatar for user Kenny Bentley
    I noticed in the case of 8/3 / 1/3 (best I can do to type a fraction division math problem on here, but basically eight thirds (8/3) divided by one third (1/3)), and likewise 8/3 / 2/3, that the denominators were both 3 in both cases, and the answer was in the same as the first numerator (8 in both cases) divided by the second numerator (1 in the first case, 2 in the second case), but a whole number and not a fraction. Is this a common pattern when dividing fractions with common denominators or are there exceptions, and would finding common denominators and dividing the numerators be one method of dividing fractions, or would it just be extra work compared to inverting and multiplying? Might be something to play around with.
    (17 votes)
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    • starky ultimate style avatar for user Paul Miller
      Yes the pattern you noticed will always work, but you are wise to suspect atet he most efficient way to divide fractions is usually to simply multiply by the reciprocal. I always remind my students to simplify the product before they multiply numerators and denominators as they are already partially factored which makes finding common factors to eliminate easier.
      (5 votes)
  • blobby green style avatar for user Kinaya Aruthur
    The video is kind of hard to understand :/
    (5 votes)
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  • aqualine sapling style avatar for user Cody Reed
    Its kinda confusing but im getting the hang of it thanks for the video!
    (4 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
    (4 votes)
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  • female robot ada style avatar for user danye
    How can I divide something like, 22 5/9 by 1/2?
    (0 votes)
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    • primosaur ultimate style avatar for user Reyna Paul
      To divide a mixed number by a fraction, you will have to convert it into an improper fraction. This means 22 5/9 would turn into 203/9.

      Dividing 203/9 by 1/2, is the same as multiplying 203/9 by 2/1.
      203/9 x 2/1 is 406/9. This can be converted back into a mixed number: 45 1/9

      Hope this helped.
      (8 votes)
  • blobby blue style avatar for user krystall
    how do you divide fraction without a number line
    (2 votes)
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  • blobby green style avatar for user Aldi Farid
    why we have to multiply it with the reciprocal when dividing is the case ?
    (3 votes)
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    • blobby blue style avatar for user Zoe
      The goal is to make the division expression look like just one number, perhaps a fraction or mixed number, but, still just one number. Multiplying by the reciprocal and multiplying by 1 result in "the product of the first fraction and the reciprocal of the second
      (1 vote)
  • leafers tree style avatar for user Anthony
    how is 6/9 changed into 2/3
    (4 votes)
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  • aqualine ultimate style avatar for user robintan
    I thought the anwer was 8/3 why is it not?
    (2 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.
      (3 votes)

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.