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Decomposing a fraction visually

Sal uses a tape diagram to decompose 7/9. Created by Sal Khan.

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  • hopper happy style avatar for user B0az0n
    How do you add numbers with different denominators (EXAMPLE: 3/9 + 2/6)?
    (10 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      You would need to convert the fractions so that they have the same denominator, and then add them by adding the values in the numerators and keeping the denominator the same. To convert a fraction into a different denominator, you have to multiply the numerator and denominator by the same number (in order to keep the actual value the same). The easiest way to convert two fractions to the same denominator is to make each denominator the least common multiple of the two previous denominators. With all that out of the way, let's see that example:
      3/9 + 2/6
      The least common multiple between 9 and 6 is 18 (you can learn how to find LCMs by using the search box in the top of any Khan Academy screen, if you don't know already). So, we multiply each fraction to get 18
      3/9 * 2/2 = 6/18
      2/6 * 3/3 = 6/18
      Now, you have two fractions with the same denominator, so you can add them as normal:
      6/18 + 6/18 = (6 + 6) / 18 = 12/18 = 2/3
      So, your final simplified answer is 2/3
      (Note: you could have made the problem lots easier by simplifying the fractions at the beginning into 1/3 and 1/3 and then just adding those, but that would defeat the purpose of learning how to add unlike denominators.)
      (43 votes)
  • duskpin tree style avatar for user Manha Mahjabeen
    what happends if you have a improper fration
    (17 votes)
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  • mr pink green style avatar for user arpolicelli
    can you do 1/9 + 6/9 - 1/9 + 1/9?
    (12 votes)
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    • leaf green style avatar for user christian33corias
      I think of it this way:
      (1/9 + 6/9) - (1/9 + 1/9)
      = (7/9) - (2/9)
      = 5/9
      So the answer would be 5/9
      I remember being taught the MDAS (Multiplication, Division, Addition, Subtraction) method (which tells you which operations you would do first in given equation). For the problem you posed, I added all terms that needed to be added then I moved on to tackle the terms that needed to be substracted....Hope this helped.
      (8 votes)
  • winston baby style avatar for user Dominic
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    (11 votes)
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  • orange juice squid orange style avatar for user Isaac
    At cant Sal use 4/9+3/9 instead of 2/9+3/9+2/9?
    (5 votes)
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    • purple pi pink style avatar for user ZeroFK
      Yes that would work too. In fact there are many more possibilities, can you find more?

      There are actually infinitely many ways to represent 7/9 (or any number really). Sal just gave 3 examples here, hoping to give the viewers some insight into fractions.
      (4 votes)
  • marcimus pink style avatar for user burrellha49
    prodigy is better!
    (6 votes)
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  • mr pink red style avatar for user chanakarn
    In Thailand, we didn't learn decomposing a fraction
    I want to know how important is it?
    (4 votes)
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  • male robot johnny style avatar for user naruto
    i want 9 prime
    (4 votes)
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  • starky tree style avatar for user cc111YT
    notice that he was having trouble while doing this. why is that? what do yall think
    (4 votes)
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  • blobby yellow style avatar for user sofia.horma
    Can you please up vote for me I got 0 upvotes
    (4 votes)
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Video transcript

So let's think about all of the different ways that we can represent 7/9. So let's just visualize 7/9. So here I have 9 equal sections. And 7/9 you could represent as 7 of those equal sections. So let me get myself a bigger thing to draw with, so that I can fill this in fast. Actually, I don't like how that looks. I'm going to use the paint brush. So here we go. So that's 1, 2, 3, 4-- you know where this is going-- 5, 6, and 7. So that's one way of representing 7/9. We already know that. That's not too interesting. But let's see if we can represent 7/9 as the sum of other fractions. So let's imagine maybe we can represent it as-- let's do it as 2/9. Let me use a different brush here. So let's represent it as 2/9. 2/9 plus-- I don't know, let's see, maybe 3/9. But that doesn't quite get us to 7/9 yet. 2/9 plus 3/9 is going to get us to 5/9. So we're going to need 2 more. So it's going to be plus another 2/9. So what would this look like? So let's just draw another grid here. So this is going to look like-- and I'll try to do it right below it, so that we can see how they match up. So we have 2/9, this 2/9 right over here. Well, we have each of these as a ninth. We have 9 equal sections. So we're going to get 1 and 2. And then we're going to add 3 more ninths. So 1, 2, 3. So we add 3/9 right over there and then 2 more ninths, 1 and 2. So notice, when I added 2/9 to 3/9 to 2/9, this equals 7/9. And we know that when we add a bunch of fractions like this that have the same denominator, we can just add the numerator. And this is why. This is 2/9 plus 3/9 times 2/9 is going to give me 7/9. Let's do this one more time. This is actually a lot of fun. So let me draw my grid again. And then let's see what we can do. So let me get my pen tool out. Let me make sure my ink isn't too thick. Well, this is fine. And let's add a couple of ninths here. So let's add first 1/9. And I'm going to draw out all the 9's in blue. And let's add 2/9. And then we could add-- I don't know, maybe we could add-- let me give some space here so we can add more. And maybe we could add 3/9. And then we could add, let's see-- actually, let me just write this out. I'm going to try to add four fractions here. So let's say add first 1/9 and see where that gets us. So 1/9 is going to get us right over here. So that's 1/9. So let's say we add 2/9 to that. I've got my little paint brush going on. So that's 1 and 2 more, 2/9. So that still doesn't get us there. This gives us a total of 3/9. 1 plus 2 is 3-- 3/9. So let's add 4/9. And I'll do that in this blue color. So 4/9. That's different enough. So let's see where this gets us. Actually, well, why not? So 4/9. And so that's going to get us 1, 2, 3, 4. So that looks like it got us all the way, because 1 plus 2 plus 4 is going to give us 7-- 7/9. So what could we put here? Well, we could say 0/9. Why not? So we could call this one right over here 0/9. And how would we visualize that? Well, we're saying none of these. No ninths right over here. So this is 1/9 plus 2/9 plus 4/9 is equal to 7/9. So these are all different ways to decompose the exact same fraction.