Welcome to a lesson
on adding signed fractions.
When adding fractions with the same, or like denominators,
we keep the denominator the same and add the numerators.
And then simplify if possible.
So for negative eight-ninths plus five-ninths
notice how we have a common denominator.
I usually rewrite this so that the negative
is in the numerator, so negative eight over nine
plus five over nine,
and again, we have a common denominator,
so the denominator stays the same.
And then we add the numerators.
Negative eight plus five is equal to negative three,
so the sum is negative three-ninths,
but this does simplify, because these two
have a common factor of three.
If we want to, we could show this as writing negative three
as negative one times three,
and nine as three times three.
So three over three simplifies to one.
And our sum is negative one-third.
So in order to add fractions, we do have to have
a common denominator.
So if we have fractions with different
or unlike denominators, the first step is going to be
to find the least common multiple of the denominators.
This is called the least common denominator, LCD.
One we determine the LCD, we'll create equivalent fractions
with that common denominator,
and then once we have the common denominator,
we will keep the denominator the same
and add the numerators and then simplify if possible.
So for negative one-fourth plus one-half,
the first step is to determine the least common multiple
of two and four, which is also the smallest number
divisible by both four and two.
Well, the LCD would be four.
If we wanted to show this, we could list multiples of four
and list multiples of two.
Notice that four is the smallest common multiple
that four and two share.
So we want to write both of these fractions
with the denominator of four.
We'll notice the first fraction already has
a denominator of four.
And then for the second fraction,
since two times two is equal to four,
we'll multiply the denominator by two
and the numerator, and multiplying by two over two
is like multiplying by one.
But now we have a common denominator of four,
so we have negative one-fourth plus two-fourths.
And now that we have a common denominator we can add.
The denominator stays the same, and since
negative one plus two is positive one,
our sum is one-fourth.
Let's take a look at some more examples.
Here we have negative six-sevenths plus five fourteenths.
So we need to find the least common multiple
of seven and fourteen.
Which is also the smallest number divisible
by both fourteen and seven.
So if we can look at this and tell the LCD is going to be
equal to fourteen, that's great.
If we can't, that's OK too.
We can list multiples of seven
and list multiples of fourteen.
So if we count by sevens, we'll have seven, fourteen,
21 and so on.
If we count by fourteens we have fourteen, 28, and so on.
We'll notice how fourteen is the smallest number
the multiples have in common.
So now we'll rewrite the first fraction
with the denominator of fourteen
so we're going to multiply both
the numerator and denominator by two.
This is going to give us negative 12/14
plus 5/14.
The denominator stays the same
and we add the numerators.
Negative twelve plus five is equal to negative seven.
And again, we're not done because seven and fourteen
do share a common factor of seven.
If we want, we can show this by writing
negative seven as negative one times seven,
fourteen as two times seven,
and there's the common factor of seven
that simplifies to one,
so our sum is negative one-half.
Next, we have negative five-sixths plus negative one-eighth.
The first step is to determine the least common multiple
of six and eight.
Which is the smallest number
divisible by both six and eight.
If we recognise this to be 24, again that's great.
If not, we can list multiples of six
and list multiples of eight.
So if we count by six we have six, twelve, eighteen,
24, 30, and so on.
Count by eights we have eight, sixteen, 24, 32, and so on.
Notice how 24 is the least common multiple,
and therefore our LCD.
So we'll have negative five-sixths plus
negative one over eight.
And since six times four is equal to 24,
multiply by four over four.
And since eight times three is equal to 24,
we'll multiply by three over three.
Notice how this is going to give us
negative 20/24.
Plus this will be negative 3/24.
We have a common denominator.
The denominator stays the same.
And we add the numerators.
Negative 20 plus negative three is negative 23.
So we have negative 23/24,
which does not simplify.
Next, we have three-tenths plus negative 1/25.
First step is to determine the least common multiple
of 10 and 25, which is the smallest number
divisible by both 10 and 25.
That would be 50.
But again, if we don't recognise this, we can list multiples
of 10 and multiples of 25.
So we'd have 10, 20, 30, 40, 50, and so on.
For 25 we'd have 25, 50, 75, and we can stop
because notice both lists contain 50.
So we're going to write both of these fractions
with the denominator of 50.
So for three-tenths, since ten times five is 50,
multiply by five over five.
And then for negative 1/25,
since 25 times two is equal to 50,
multiply the top and bottom by two.
So now we have 15/50
plus negative 2/50.
Denominator stays the same,
and we have 15 plus negative 2, which is equal to 13,
which doesn't simplify, and therefore this is our sum.
In the last example, we have negative three
plus negative 5/12.
To write negative three in fraction form,
we can put this over one.
So if our denominator is one and twelve,
the least common denominator is going to be twelve.
So for negative three over one,
since one times twelve is equal to twelve,
we'll multiply the top and bottom by twelve.
The second fraction already has a denominator of twelve.
So give us negative 36/12
plus negative 5/12.
And again, the denominator is twelve,
so it stays twelve and then we add the numerators.
Negative 36 plus negative five is equal to negative 41.
And again, there are no common factors here,
therefore this is our sum in simplified form.
There are other ways to find the LCD,
but listing multiples like this will always work.
I hope you found this helpful.