We want to add or subtract the fractions,
and write the answer in simplified form.
The most important thing to remember
about adding or subtracting fractions,
is that we must obtain a common denominator,
before we add or subtract.
Hopefully we can find the least common denominator,
but we must have at least a common denominator,
because by doing this,
we're adding or subtracting pieces or partitions
of the same size.
Let's look our model before we begin.
Let's consider 1/2 plus 1/4.
Notice how here, the least common denominator would be four,
which means you'd have to multiply 1/2 by two over two,
to obtain an equivalent fraction with a denominator of four.
So instead of 1/2 plus 1/4, we have 2/4 plus 1/4.
So notice how, because our denominators are the same,
we'll be adding pieces of the same size.
Notice here's the model for 2/4,
and here's the model for 1/4.
Now, because the partitions or pieces are the same size,
we can see we'll have a total of three pieces,
where each piece is 1/4, giving us a sum of 3/4.
So by determining a common denominator,
we're making sure that we add or subtract
pieces of the same size.
Now, let's also look at a common error.
A common error when adding fractions,
would be to add the numerators and add the denominators.
Using our models, we can see why this does not make sense.
Here's our model for 1/2, here's our model for 1/4.
If we add these two pieces together,
there's no way we can have two pieces,
each the size of 1/6, giving us a sum of 2/6.
This method does not work.
We must obtain a common denominator,
before we add or subtract fractions.
So looking at our first example here, we have 1/4 plus 5/8.
So ideally we want to find the least common denominator,
which would be the least common multiple of four and eight,
which would be eight.
So we want to write 1/4 as an equivalent fraction,
with a denominator of eight.
We can do this by multiplying 1/4 by two over two.
So now this would be 2/8, plus 5/8.
Now that our denominator is the same,
the denominator's going to stay eight,
and the numerator is going to be two plus five or seven.
Our sum is 7/8, which does not simplify.
Next we have 5/6 minus 5/12.
Again, we're looking for the least common denominator,
which is the least common multiple of six and 12,
which would be 12.
So we're going to write 5/6 as an equivalent fraction
with a denominator of 12,
so we'll multiply 5/6 by two over two.
So this will give us, notice 10/12 minus 5/12.
Notice because we have a common denominator,
the partitions or pieces are the same size,
so it'll stay twelfths, and the numerator is going to be
10 minus five, which equals five.
This does not simplify, and therefore,
this is our difference 5/12.
Next, we have 5/7 plus 3/5.
Notice here, both five and seven are prime,
and therefore, the least common denominator
is going to be their product, seven times five equals 35.
We're going to write both fractions, with a denominator of 35.
So we'll multiply 5/7 by five over five.
And we'll multiply 3/5 by seven over seven.
So we have 25/35 plus 21/35.
So our denominator's going to stay 35,
and the numerator is going to be 25 plus 21,
which equals 46.
So notice here, we have an improper fraction.
It does not simplify, because 46 and 25
don't have any common factors other than one.
But notice how we are asked to enter a reduced fraction
or mixed number in simplified form,
plus also convert this to a mixed number.
This fraction means 46 divided by 35, so 46 divided by 35.
There's one, 35, and 46, let me subtract.
And we have a remainder of 11,
which means 46/35 is equal to one whole, and 11/35.
We have the remainder over the divisor.
Let's go ahead and enter our mixed number, one and 11/35.
And now for our last example, we have 1/3 minus 1/4.
So we're looking for the least common multiple
of three and four, which is our least common denominator,
which would be 12, so multiply 1/3 by four over four,
and we'll multiply 1/4 by three over three.
So we have 4/12 minus 3/12,
so our denominator stays 12, and the numerator
is four minus three, which equals one,
so our difference is 1/12.
And that's going to do it for these examples.
I hope you found this helpful.