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### Course: 7th grade (Illustrative Mathematics) > Unit 5

Lesson 3: Lesson 3: Changing elevation# 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.

## Want to join the conversation?

- How can I do this without a number line? Please help!(65 votes)
- 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?(17 votes)

- Seems a bit much for one equation, is there any simpler/faster way to do it?(17 votes)
- 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`

(34 votes)

- 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(8 votes)
- 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)

- this is so confusing and there is so much to remember(15 votes)
- 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(7 votes)- 6-2 is not the same as -12/6+1/6 but -12/6+1/6 is -11/6(7 votes)

- While in the video type the word awesome, and the time bar turns rainbow, it is really cool.(9 votes)
- Is it the same principle but with fractions?🤔(8 votes)
- what are the steps(1 vote)
- Well, the first step would be to simplify the equation. For example, if I had the problem

-6 3/7 - -(2 7/9) - 8 3/10

then I could simplify it to:

-6 3/7 + 2 2/5 - 8 3/10

Then, you add the fractions together, and the first step for that is to find the common denominator. So, the common denominator of 7, 5, and 10, (the denominators), would be 70, and change the denominators to 70, and multiply the numerators by the number you multiplied the denominator by.

Here's an example, using our above equation;

-6 3/7 + 2 2/5 - 8 3/10

So, we change the denominators to 70, so our equation would be:

-6 3/70 + 2 2/50 - 8 3/70

But now our numerator wouldn't look right, so we have to multiply the numerators by the number we multiplied the denominator by.

So, for -6 3/70 to go from 7 to 70, we would have to multiply by 10, so we multiply 3 by to to get 30, so from:

-6 3/7 to

-6 30/70

The actual value of the fraction stays the same, though this format makes it easier to add/subtract.

So, if we repeat this process, our equation will be

-6 30/70 + 2 28/70 - 8 21/70

Now we do the actual addition. Let's focus on the whole numbers first.

So -6 + 2 - 8

What is 6 to the**left**of 2? -4! And what is 8 to the left of -4? -12! You can use a number line to help you out. So, the sum of all the whole numbers is 12.

Now let's do the fractions:

-30/70 + 28/70 - 21/70 = -23/70

Make sure you include all the minuses and plusses that were in the equation.

The last step is to add the two products.

-12 + -23/70 = -12 23/70

You then simplify, but this specific fraction is already simplified, so our final answer was:

-12 23/70

I know this looks long and like a bore to read through, but I still hope this helps you! Have a good day!😁(12 votes)

- Do they have negative decimals, so can you make the -2 1/6 -2.6 repeating?(4 votes)
- You can definitely have negative decimals.(2 votes)

## 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.