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Adding fractions with like denominators

Sal adds 3/15+7/15. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • mr pants teal style avatar for user KXZivkovic
    Does GCD mean greatest common divider. If so than how is it any different from LCM?
    (203 votes)
    • leaf green style avatar for user Elijah Lape
      Yes, GCD means Greatest Common Divisor. Another way to think of it is the Greatest Common Factor. When comparing 2 or more numbers, the GCD/GCF is the LARGEST (i.e. greatest) factor that they both share. For example, 12 and 16. The factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 16 are 1, 2, 4, 8, 16. If you look at both lists, you'll notice that both share the numbers 1, 2 and 4 but the largest of these shared COMMON factors is 4 so the GCD or GCF is 4. You'll notice that this is smaller than both 12 and 16.

      LCM is the Lowest/Least Common Multiple. MULTIPLES on the other hand are larger or equal to the numbers you are comparing. Take 3 and 8. The multiples of 3 is just the 3 times table (3, 6, 9, 12, etc.) and the multiples of 8 are the 8 times table (8, 16, 24, 32, and so on). If you continued each table you would eventually find that each table shares some numbers. The SMALLEST (i.e. least/lowest) of these that are shared is the LCM. In this case, continuing 3's multiples: 15, 18, 21, 24, 27... etc. You can see that 8 and 3 can both multiply to 24. So the LCM is 24. Using the previous example of 12 and 16, the LCM is 48 since 12 x 4 = 48 and 16 x 3 = 48.
      (65 votes)
  • spunky sam blue style avatar for user Voltaire
    Why do the denominators have to be the same?
    (12 votes)
    • leafers seedling style avatar for user kristin.willard
      The denominators have to be the same so you can add the numerators together without worrying about the denominators being different sizes, because that affects the value of the fraction.

      It's like trying to count the number of pieces you can get out of different cakes. If the cakes are different sizes, it wouldn't be fair because some people would get larger pieces than others. By making cakes that are the same size and then counting the number of pieces, you can be sure that everyone is getting a fair amount of cake.

      In order to add fractions correctly, the "cakes" need to be the same size, which is why the denominators (bottom numbers in the fractions you are adding) need to be the same value.
      (30 votes)
  • leaf blue style avatar for user Programming Crazy
    Why don't you add the denominator?
    (0 votes)
    • hopper cool style avatar for user 15tkostolansky
      If you added the denominators, the whole answer would not make sense. If you added 1/2+1/2 and got 2/4, that is wrong, because two halves equal a whole, but 2/4 is just another fraction equivalent to 1/2. In fraction adding, you are just adding the parts (numerators), not the the amount of all the pieces (denominators). Think about it this way:
      You have 3 of the 5 pieces left of pumpkin pie and 4 of the 5 pieces of apple pie, and you want to add them. You add 3+4 and get 7 pieces. Now you have 7/5 of a pie. This is because only five pieces fit in a pie and you have 7 pieces of pie left.
      Hope this helps!
      (6 votes)
  • male robot donald style avatar for user Jason Weber 75
    at , Sal said that if the denominator is the same, you just add the numerator and then the denominator stays the same? I just dont get it. Is it the same on the problems without the same denominators?
    (64 votes)
    • piceratops ultimate style avatar for user hmmm
      If the denominators are different, then the formula will not be the same. Let's say you have this problem:
      1/3 + 1/3.
      The denominators are the same, so they will not change. You simply add the numerators and keep the denominator.
      1/3 + 1/3 = 2/3

      However, if you have this problem:
      You have to make the denominators the same. A way to do this is to divide the larger denominator by the smaller denominator to find the GCD (Greatest Common Divisor):
      Then multiply the numerator and denominator of 1/2 by 2 (our GCD):
      1 x 2 = 2
      2 x 2 = 4
      The denominators are now the same, so let's add:
      2/4 + 1/4 = 3/4
      1/2 + 1/4 = 3/4!
      I hope this helped!
      (36 votes)
  • male robot hal style avatar for user Novantix
    What if i have 6/10 - 8/10 how do I solve that? Because the number is smaller than the other and I dont have a mixed fraction to borrow from.

    Please help :)
    (9 votes)
  • primosaur ultimate style avatar for user Bitty Bob Jones
    What if your sum goes over the denominator? Say, 7/14 + 9/14. How do you deal with that?
    (12 votes)
  • aqualine tree style avatar for user magicprettygirl04
    How and why does this work ?
    (4 votes)
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    • winston default style avatar for user Merlin Wang
      The top number represents the amount you have, and the bottom number represents the total needed to form a whole item. so if you have 4 slices of pizza on one plate and another 4 slices of pizza on another plate, and 8 slices form a whole pizza, you now know the denominator is 8, because that is the amount you need to make a whole pizza. Now, if you take the numerator (The actual amount) and add it, you aren't changing how many slices you need to make a pizza, but how many you actually have. When you put all slices together on a plate, you have a whole, or 8/8 pizza.
      (6 votes)
  • purple pi teal style avatar for user Pi
    Does GCF mean greatest common factor ?
    (4 votes)
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  • piceratops seedling style avatar for user Ablyss
    Why do you divide the fraction by 5 though?
    (3 votes)
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  • purple pi pink style avatar for user myzack001
    when will you add fractions in real life?
    (1 vote)
    Default Khan Academy avatar avatar for user

Video transcript

So we're asked to add 3/15 plus 7/15, and then simplify the answer. So just the process when you add fractions is if they already-- well, first of all, if they're not mixed numbers, and neither of these are, and if they have the same denominator. In this example, the denominators are already the same. The denominator is 15. So if you add these two fractions, your sum is going to have the same denominator, 15, and your numerator is just going to be the sum of the numerator, so it's going to be 3 plus 7, or it's going to be equal to 10/15. Now, if we wanted to simplify this, we'd look for the greatest common factor in both the 10 and the 15, and as far as I can tell, 5 is the largest number that goes into both of them. So divide the 10 by 5 and you divide the 15 by 5, and you get-- 10 divided by 5 is 2 and 15 divided by 5 is 3. You get 2/3. Now, to understand why this works, let's draw it out. Let's split something up into 15 sections. So let me split it up into 15 sections. Let me see how well I can do this. Well, actually, even a better way, an easier way might be to draw circles. So let me do the 15 sections. So let me draw. So that is one section right over there. That is one section and then if I copy and paste it, that is a second section, and then a third section, fourth section, and then we have a fifth section. Let me copy and paste this whole thing. So that's five sections right there. Let me copy and then paste that. So that is 10 sections, and then let me do it one more time. So that is 15 sections. So you can imagine this whole thing is like a candy bar or something, and we have now split it up into 15 sections. Now, what is 3/15? Well, it's going to be 3 of the 15 sections. So 3/15 is going to be one, two, three: 3/15. Now, to that, were adding 7 of the 1/15 sections, or 7 of the sections. So we're adding 7 of those to it. So that's one, two, three, four, five, six, seven. And you see now, if you take the orange and the blue, you get one, two, three, four, five, six, seven, eight, nine, ten of the sections, or 10 of the 15 sections. And then to see why this is the same thing as 2/3, you can just split this candy bar into thirds, so each third would have five sections in it. So let's do that. One, two, three, four, five, so that is 1/3 right there. One, two, three, four, five, that is another third right there. And notice, when you do it like this, we have filled out exactly two-- one, two-- of the thirds. This is the third third, but that's not filled in. So 10/15 is the same thing as 2/3.