Adding & subtracting negative fractions
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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.