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Understanding fractions as division

In this video, we learn about the relationship between multiplication and division. Let's watch how they can undo each other. We can understand this concept with whole numbers and fractions. Created by Sal Khan.

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  • primosaur sapling style avatar for user BraxtonM
    so we can do division or multiplecarion
    (30 votes)
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  • aqualine sapling style avatar for user Yes
    If you still don't understand how to divide fractions. I will show you. There are 3 steps.
    So say you were doing 1/3 divided by 6 or 6 divided by a third.
    We will do 1/3 divided by 6 first.
    Step 1: Keep the number 1/3
    Step 2: Switch the division symbol to multiplication
    Step 3: Find the reciprocal of 6. And its 1/6 so 1/3 times 1/6 is 1/18. Sometimes you need to simplify.
    Now 6 divided by 1/3
    Keep 6.
    Turn to muliplication
    Switch 1/3 to 3/1
    its easier as 6/1x3/1 and that's 18/1 which can be reduced to 18
    Hope this helped
    (24 votes)
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  • piceratops seed style avatar for user Willams Evan
    So multiplying fractions is the almost same as dividing them?
    (8 votes)
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  • leaf green style avatar for user Areau
    Ok so question! I want to find (3/4) / (7/8) = x. Now the initial 3/4 = 0.75 and 7/8 = 0.875 then take 0.75 / 0.875 = 0.857 (rounding) then it becomes 857/1000 <- what is the best way of finding out 857/1000 in its simplest form aside from rounding.


    Another question so (3/4) / 7 = x. becomes 0.6 / 7 = 0.086 (rounding) 86/1000 simplified = 43/500? (Then once again how do I know I have found the lowest simplified form)
    (7 votes)
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    • hopper cool style avatar for user McCringe
      you're really overcomplicating it, just mutiply their denomiators by eachother to get a common whole. whatever you had to mutiply by to get that number from the previous denomiator is how much you mutiply numerator. now you have equal fractions, and you can divide or mutiply them easily.
      (6 votes)
  • starky sapling style avatar for user Yocelynh2010
    Any tips? I would love that please! I still cant do it myself.
    (5 votes)
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    • aqualine ultimate style avatar for user AzhinHacker
      Swap the numerator and the denominator, (the top and bottom numbers in a fraction) on the second fraction. WHATEVER YOU DO, DO NOT DO IT FOR THE FIRST FRACTION. For example, 1/2 x 3/9, so it would be re-written as 1/2 x 9/3, and then you just multiply it, which would be 9/6, and then, you divide the numerator by the denominator, so just think, "how many times can 6 go into 9? And how much is left over?" So, if you were to do that, the answer would be 1 3/9, which, by the way, was you second fraction that you swapped. And, if you know how to, then you would simplify, and your final answer would be 1 1/3. (Oh, and by the way, the word for swapping the numerator and the denominator is called recipricals. So if you ever see that word in a test or a math question, then you know what it is.
      (5 votes)
  • blobby green style avatar for user ally.redden
    Sam'' it actually makes a lot of sense with fractions'' Me '' Bro what is 2 times 4?''
    (7 votes)
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  • blobby green style avatar for user noah.gill218
    that was boring and easy to understand
    (5 votes)
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  • hopper happy style avatar for user capybara72nationalcow
    if you have infinity forths how would you divide it
    (4 votes)
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  • winston baby style avatar for user ﮩـﮩﮩ٨ـ🫀ﮩ٨ـﮩﮩ٨ـ
    sal khan on top
    (4 votes)
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  • blobby green style avatar for user johncena1388
    How do you divide 1 into 3? I know it will equal a fraction but I don't know how it becomes a fraction. That is the only part that I don't understand in the video.
    (2 votes)
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Video transcript

When we were first exposed to multiplication and division, we saw that they had an inverse relationship. Or another way of thinking about it is that they can undo each other. So for example, if I had 2 times 4, one interpretation of this is I could have four groups of 2. So that is one group of 2, two groups of 2, three groups of 2, and four groups of 2. And we learned many, many videos ago that this, of course, is going to be equal to 8. Well, we could express a very similar idea with division. We could start with 8 things. So let's start with one, two, three, four, five, six, seven, eight things. So now we're going to start with the 8. And we could say, well, let's try to divide that into four groups, four equal groups. Well, that's one equal group, two equal groups, three equal groups, and four equal groups. And we see when we start with 8 divide it into four equal groups, each group is going to have 2 objects in it. So you probably see the relationship. 2 times 4 is 8. 8 divided by 4 is 2. And actually, if we did 8 divided by 2, we would get 4. And this is generally true. If I have something times something else is equal to whatever their product is, if you take the product and divide by one of those two numbers, you'll get the other one. And that idea applies to fractions. It actually makes a lot of sense with fractions. So for example, let's say that we started off with 1/3 and we wanted to multiply that times 3. Well, there's a couple of ways we could visualize it. Actually, let me just draw a diagram here. So let's say that this block represents a whole, and let me shade in a 1/3 of it. So that's 1/3. We're going to multiply by 3. So we're going to have 3 of these 1/3's. Or another way of thinking about it, it's going to be 1/3 plus another 1/3 plus another 1/3. That's our first 1/3, our second 1/3, and our third 1/3. And we get the whole. This is 3/3, or 1. So this is going to be equal to 1. So you use the exact same idea. If 1/3 times 3 is equal to 1, then that means that 1 divided by 3 must be equal to 1/3. And this comes straight out of how we first even thought about fractions. The first way that we ever thought about fractions was, well, let's start with a whole. And that whole would be our 1. And let's divide it into 3 equal sections, the same way that we divided this 8 into 4 equal groups. So if you divide this into 3 equal sections, the size of each of those sections is going to be exactly 1/3. Now, this leads to an interesting question that might be popping in your brain. Notice, we have 1 is the numerator, 3 is the denominator, and we just said that this is equal to the numerator divided by the denominator. 1 over 3 is the same thing as 1 divided by 3. Is this always true for a fraction? Well, let's just do the same thought experiment, but let's do it with a different fraction. Let's take 3/4 and multiply it by 4. So multiply it by 4. So once again, let's see if I could draw 1/4 here. Let me do this in a new color. So let's say that this block right over here is a whole. We'll divide it into four equal sections. So now I've divided it into fourths. And let me copy and paste it so I can use it multiple times. So copy. All right. Now, 3/4, that's going to be-- we can assume-- I didn't draw it perfectly. Actually, I could draw it a little bit better than that just to make the four equal sections actually look equal. So that looks like a little bit better of a job. I'm trying to make them four equal sections. Let me copy that one. So let me use it for later. Now, 3/4. This is four equal sections, and 3/4 represents three of them-- one, two, three. But now we're going to multiply it by 4. So we're going to have 3/4 four times. So we're going to need some more wholes here. So let's throw in another whole. So this is one 3/4. Now let me do the next 3/4 in another color. So that's a 1/4, that's a second 1/4, that's a third 1/4. That's another 3/4. And now let's do-- so we've done two 3/4 just now. Let me make it clear. This is the first 3/4, and then this plus this is the second 3/4. Now let's do a third 3/4. And we're going to have to use another whole right over here. And I will do that in this color. So my third 3/4, so here's a 1/4, here's my second 1/4, here's a third 1/4. So in green, I have another 3/4. And now we need four 3/4. So let's do that in a color I have not used yet, maybe white. So that's a 1/4, that's two 1/4, and that is three 1/4. So notice, now I have now I have one 3/4, two 3/4, three 3/4, and four 3/4. And what did I do when I got those four 3/4? Well, it's pretty clear. This is turned into 3 wholes. So this is equal to 3 wholes. Well, if 3/4 times 4 is equal to 3, that means that 3 divided by 4 is equal to 3/4. So the same idea again. 3 over 4 is the same thing as 3 divided by 4. And in general, this is true. The fraction symbol here can be interpreted as division. And looking at this diagram right here, it made complete sense. If you started with 3 wholes, and you want to divide it into 4 equal groups, one group, two groups, three groups, four groups, each group is going to have 3/4 in it.