Adding fractions with different signs

Find the sum 3 1/8 + 3/4 + ( -2 1/6)

Let's just do the first part first,

This is pretty straightforward.

We have two positive numbers, let me draw a

number line , we are going to start with 3 1/8,

this is 0, then we have 1, 2, 3 and then you have 4.

3 1/8 is going to be right about there

so let me just draw its absolute value

3 1/8, is going to be 3 1/8ths to the right of 0

So it is going to be exactly that distance

from 0 to the right.

The length of this arrow, you could view it as 3 1/8th.

Now, whenever I like to deal with fractions

especially when I have different denominators

i like to deal them with improper fractions,

it makes the addition, and the subtraction,

and actually the multiplication and divison a lot easier.

So 3 1/8 is the same thing as

8 times 3 is 24 plus one is 25 over 8.

which is the same thing as 3 1/8.

Another way to think about it,

3 is 24 over 8,

and then add 1/8 to that,

so that you get 25/8.

So this is our starting point.

To that we are going to add 3/4,

we are going to move another 3/4

to the right, this length of this is 3/4

So plus 3/4.

Where does this put us,

so both of these are positive intergers,

so we can just add them,

we just have to find the like denominators

we have 25/8 + 3/4. That's the same thing as,

we have to find the 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 multiplied by 2,

so we have to multiply 3 by 2 as well.

So we get 6,

so 3/4 is the same thing as 6/8.

So we have 25/8, and we're adding 6/8 to that,

that gives us 25 plus 6 is

31, over 8.

So this number right over here is 31/8

And it makes sense, because 32/8 will be 4,

so it should be a litte bit less than 4

So this number right over here is 31/8

or the length of this arrow,

the absolute value of that number

is 31/8.

If you want to write it as a mixed number, it would be

3 7/8.

Now to that we want to add a -2 1/6.

So we're going to add a negative number.

So think about what -2 1/6 is going to be like.

Let me do this in a new color, do it in pink.

-2 1/6. So we're gonna subtract, or,

I guess we should say we are going to add a -1,

add a -2, and then a -1/6.

So -2 1/6, we can literally draw it like this.

-2 1/6, we can draw it with an arrow

that looks something like that,

so this is -2 1/6.

There are a couple of ways to think about it,

we could just say,

when you add this arrow, this thing is moving to the left.

We could put it over here

and you would get straight to -2 1/6.

But we're adding this -2 1/6,

it is same thing as subtracting a 2 1/6;

we're moving 2 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.

It is actually going to be to the right,

so it's not only going to be its absolute value,

the 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 going to be the difference of 31/8 and 2 1/6.

And it's a positive difference,

because we're dealing with a positive number.

So we just take 31/8, and

from that, we will subtract 2 1/6.

So this orange value is going to be 31/8

minus 2 1/6.

So 2 1/6 is the same thing as

6 times 2 is 12 plus 1 is 13.

-13/6, and this is equal to --

the common denominator over here -

once again we're going to get

and it looks like 24 will be the common denominator.

This is the 31/8, and this is the 2 1/6.

So 31/8, over 24, you have to multiply by 3

to get 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,

I'll do that in another color,

so you 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,

3 minus 2 is 1, 9 minus 5 is 4.

So it is 41/24, positive.

You can see that here,

just by looking at the number line.

This right here is 41/24, and it should be

a little bit less than 2,

because 2 would be 48/24.

So this would be 48/24, and it makes sense

because we're are a little bit less than that.