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Adding fractions with different signs

Use a number line to add fractions with different signs.  Created by Sal Khan and Monterey Institute for Technology and Education.

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  • boggle blue style avatar for user Zainab Hady
    How can I do this without a number line? Please help!
    (55 votes)
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    • aqualine ultimate style avatar for user Castro, Jocelyn
      Well if you want to do it without a number line then you just have to do the math. What I mean is that you have to first know how to add and subtract fractions and you have to learn how to learn how to add and subtract mixed fractions, also you have to learn how to add and subtract integers. So yeah it takes a long time to do this kind of math to learn how solve a problem without a number line. Understand?
      (15 votes)
  • aqualine ultimate style avatar for user Settles, Zipporah
    Seems a bit much for one equation, is there any simpler/faster way to do it?
    (13 votes)
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    • leafers ultimate style avatar for user Matthew
      Yes, you can do just do the math without the visualization alone, the number line is just for better understanding. It is a bit complicated, but it is the fastest and most simple way to handle the question if you understand the math to it.

      3 1/8 + 3/4 +(-2 1/6) = x
      Let's solve for x.

      x = 3 1/8 + 3/4 - 2 1/6 [delete the ()]
      x = 3 1/8 + 6/8 - 2 1/6 [adjust the fractions for addition]
      x = 25/8 + 6/8 - 2 1/6 [convert into improper fraction]
      x = 31/8 - 2 1/6 [do the addition]
      x = 31/8 - 13/6 [convert into improper fraction]
      x = 93/24 - 52/24 [adjust the fractions for subtraction]
      x = 41/24

      or...

      x = 1 17/24
      (24 votes)
  • male robot johnny style avatar for user stephen.f.howe
    When I worked through this problem, I left them as mixed numbers and didn’t convert them to improper fractions. I came to the same answer (41/24 or 1&17/24). Did you convert them for a reason other than preference? Thanks
    (10 votes)
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    • piceratops ultimate style avatar for user Logan M.
      Yes, the "short way" of doing this problem is just working on the fractions of the mixed numbers (1/8+3/4-1/6) and subtracting the numbers (3-2). The reason this method works is because, in reality, 1&17/24 is the same as 1+17/24 (and this goes for all mixed numbers). Sal converted the mixed numbers mostly because of preference, but it is easier to convert to improper fractions when the problem would result in a negative fraction.
      For example:
      3&1/8 - 3/4 -2&1/6
      See, now the fractions added up would result in a negative fraction (-19/24) and then you would still have to subtract 19/24 from 1. So, you would have to convert 1 to 24/24 and subtract. By converting all the mixed numbers into improper fractions, the sum of all the improper fractions is the final answer.
      In conclusion, both methods work (Sal's method is the one taught in schools, normally) and, depending on the question (whether the fractions of the mixed numbers come out negative), one can work faster than the other.
      (22 votes)
  • blobby green style avatar for user Alan Yves
    I tried to follow the lesson and I came to -11/6 instead of -13/6.
    Could we get a clarification on this or I missed a step?
    Look forward to her from you guys! example 6. -2 = -12/6 + 1/6= -11/6
    (9 votes)
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  • duskpin ultimate style avatar for user *Kayla*
    this is so confusing and there is so much to remember
    (11 votes)
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  • leafers sapling style avatar for user SVstudent
    Is it the same principle but with fractions?🤔
    (9 votes)
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  • primosaur seed style avatar for user Walker Bryson
    I think the number line was so confusing and was wondering if there where any other way you would beable to teach this without a number line. Thank you.
    (6 votes)
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  • blobby green style avatar for user ecruz76
    I dont understand
    (4 votes)
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  • leaf grey style avatar for user jeffery
    Why would you subtract 2 1/6 from 31/8? Why wouldn't you add? where did the addition symbol go? This has been confusing me and making me fail my exercises...
    (2 votes)
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  • leaf grey style avatar for user Jlfrear
    I suck at math. Is there another way to simplify
    (5 votes)
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Video transcript

Find the sum 3 and 1/8 plus 3/4 plus negative 2 and 1/6. Let's just do the first part first. It's pretty straightforward. We have two positive numbers. Let me draw a number line. So let me draw a number line. And I'll try to focus in. So we're going to start at 3 and 1/8. So let's make this 0. So you have 1, 2, 3, and then you have 4. 3 and 1/8 is going to be right about there. So let me just draw its absolute value. So this 3 and 1/8 is going to be 3 and 1/8 to the right of 0. So it's going to be exactly that distance from 0 to the right. So this right here, the length of this arrow, you could view it as 3 and 1/8. Now whenever I like to deal with fractions, especially when they have different denominators and all of that, I like to deal with them as improper fractions. It makes the addition and the subtraction, and, actually, the multiplication and the division a lot easier. So 3 and 1/8 is the same thing as 8 times 3 is 24, plus 1 is 25 over 8. So this is 25 over 8, which is the same thing as 3 and 1/8. Another way to think about it, 3 is 24 over 8. And then you add 1/8 to that, so you get 25 over 8. So this is our starting point. Now to that, we are going to add 3/4. We are going to add 3/4. So we're going to move another 3/4. We are going to move another 3/4. It's hard drawing these arrows. We're going to move another 3/4 to the right. So this right here, the length of this that we're moving to the right is 3/4. So plus 3/4. Now where does this put us? Well, both of these are positive integers. So we can just add them. We just have to find a like denominator. So we have 25 over 8. We have 25 over 8 plus 3/4. That's the same thing as we need to find a common denominator here. The common denominator, or the least common multiple of 4 and 8 is 8. So it's going to be something over 8. To get from 4 to 8, we multiply by 2. So we have to multiply 3 by 2 as well. So you get 6. So 3/4 is the same thing as 6/8. If we have 25/8 and we're adding 6/8 to that, that gives us 25 plus 6 is 31/8. So this number right over here, this number right over here, is 31/8. And it makes sense because 32/8 would be 4. So it should be a little bit less than four. So this number right over here is 31/8. Or the length of this arrow, the absolute value of that number, is 31/8, a little bit less than 4. If you wanted to write that as a mixed number, it would be what? It would be 3 and 7/8. So that's that right over here. This is 31/8. That's that part right over there. Now to that, we want to add a negative 2 and 1/6. So we're going to add a negative number. So think about what negative 2 and 1/6 is going to be like. So let me do this in a new color, do it in pink. Negative 2 and 1/6. So we're going to subtract, or I guess we're going to say we're going to add a negative 1. We're going to add a negative 2 and then a negative 1/6. So let me draw. So negative 2 and 1/6, we could literally draw like this. Negative 2 and 1/6 we could draw with an arrow that looks something like that. So this is negative 2 and 1/6. Now, there's a couple ways to think about it. You could just say, hey, look, when you add this arrow, this thing that's moving to the left-- we could put it over here, and you would get straight to negative 2 and 1/6. But we're adding this negative 2 and 1/6. It's the same thing as subtracting a positive 2 and 1/6. We're moving 2 and 1/6 to the left. And we're going to end up with a number whose absolute value is going to look something like that. And it's actually going to be to the right. So it's not going to only be its absolute value. Well, its absolute value is going to be the number since it's going to be a positive number. So let's just think about what it is. This value right here, which is going to be the answer to our problem, is just going to be the difference of 31/8 and 2 and 1/6. And it's the positive difference because we're dealing with a positive number. So we just take 31/8. And from that, we will subtract 2 and 1/6. So let's do this. So this orange value is going to be 31/8 minus 2 and 1/6. So 2 and 1/6 is the same thing as 6 times 2 is 12 plus 1 is 13. Minus 13/6. And this is equal to, once again, we need to get a common denominator over here. And it looks like 24 will be the common denominator, 24. And let me make it very clear. This is the 31/8. And this is the 2 and 1/6. This right here is the 2 and 1/6. So 31/8 over 24. You have to multiply by 3 to get to the 24 over here. So we multiply by 3 on the 31. That gives us 93. And then to go from 6 to 24, you have to multiply by 4. We do that in another color. You have to multiply it by 4, so we have to multiply by 4 up here as well. So 4 times 13, let's see. 4 times 10 is 40. 4 times 3 is 12. So that's 52. So this is going to be equal to 93 minus 52 over 24. And that is-- so 93 minus 52. 3 minus 2 is 1. 9 minus 5 is 4. So it is 41/24 and positive. And you can see that here just by looking at the number line. This right here is 41 over 24. And it should be a little bit less than 2 because 2 would be 48 over 24. So this would be 48 over 24. And it makes sense that we're a little bit less than that.