Mixed Numbers

A mixed number is a number made up of a whole number and a fraction. It means that you have
1 (or more) wholes, and a part (the fraction). A mixed number looks like this:

or like this: 1 3/4, depending on how you write it. There are lots of things you
can do with mixed numbers; you can add, subtract, multiply, and divide them, just
like you can with whole numbers or fractions. There are several steps you have to
remember in order to work with mixed numbers correctly.

  • When adding or subtracting mixed numbers, you must align the columns. That means
    that whole numbers should line up with whole numbers, and fractions should line
    up with fractions. An aligned problem looks like this:

  • In order to add or subtract fractions, you must have common denominators. The same
    is true for mixed numbers—you must find a common denominator between the fractions
    before you can add or subtract.
  • In order to multiply or divide mixed numbers, they must be in the form of improper
    fractions. This means that before you multiply or divide them, you must convert
    them into improper fractions. For more help with improper fractions, see Improper Fractions. This is one time when
    it’s alright to have improper fractions.
  • When you have an answer that is a mixed number, always check to make sure that the
    fraction is reduced all the way! The answer is wrong unless the fraction is reduced
    (simplified).
  • When borrowing, you need to remember that 1 has a fraction form that is anything
    over itself. For example, if you are working with fractions with a denominator of
    15, you would need to write 1 as a fraction like this: 15/15.

Adding Mixed Numbers

Adding mixed numbers may seem hard, but it’s just putting together multiple steps
of what you already know. For example, you might have a problem that looks like
this:

This is a basic mixed number addition problem. First you add the fractions together.
Then, add the whole numbers together. Your answer to a mixed number addition problem
will most likely also be a mixed number. Your solution to this problem would look
like this:

Then, you would add the whole numbers together, like this: 5 + 3 = 8.

Thus, your answer is 8 3/4.

Now, what happens if you have a more complicated problem, something like this?

You’ll notice that we do not have common denominators, but in order to add these
two numbers together, we need common denominators! Therefore, step one is to find
common denominators, like this (for a more in-depth explanation of this, see How to Find Common Denominators):

4: 4, 8, 12, 16, 20

8: 8, 16, 24, 32, 40

After you list your multiples, you will see that the least common multiple between
4 and 8 is 8. So, we’ll use 8 as our common denominator. That means we only have
to expand the first fraction to give it a denominator of 8). That would look like
this:

Now that we have common denominators, we can add the fractions together, like this:

And then add the whole numbers together, 8+1=9

Thus, your answer is 9 7/8.

Adding Mixed Numbers Resulting in an Improper Fraction

Sometimes, you run across an improper fraction when you do your addition. For example,
take the last problem but instead of adding 1/8 you’re going to add 3/8. It would
look like this:

Once you change the denominators so that they’re both 8, it would look like this:

Now, let’s start with the fraction addition. Your problem then looks like this:

But when you add 6+3 together, you get 9, which makes your fraction 9/8, an improper
fraction! Leave it as an improper fraction for just a second, while you add the
whole numbers together. The whole numbers (8+1) = 9, so now we have 9 9/8…

but we can’t leave it like this! We know that our numerator cannot be bigger than
our denominator, so we have to change the improper fraction into a mixed number
(for extra help with this, see
Improper Fractions
).

To do this, we know that 9 divided by 8 = 1 r 1, so we take our remainder (1) and
put it over 8 (our denominator) so our new fraction is 1/8. We take the whole number
we got during division (1) and add it to our whole number from the addition (9)
and we have 9 + 1 = 10.

Thus, our final answer is 10 1/8.

Subtracting Mixed Numbers

Subtracting mixed numbers is very similar to adding mixed numbers. For instance,
you may see a problem that looks like this:

Notice that we have common denominators, so we don’t have to worry about expanding
either of our fractions. To solve this problem, first subtract the fractions. That
would look like this:

Then subtract your whole numbers: 5 – 2 = 3.

Thus, our answer is 3 2/5.

Sometimes, you might see a mixed number subtraction problem with uncommon denominators.
Don’t worry, you just find a common denominator and do the subtraction as usual,
like this:

First, we have to find a common denominator. We realize the least common multiple
between 5 and 3 is 15, so our common denominator is 15. We expand both fractions
so that they have a denominator of 15, which would look like this:

Now both of our fractions have denominators of 15, so we can subtract them. The
subtraction looks like this:

After subtracting the fractions, move on and subtract the whole numbers: 5 – 4 =
1.

Thus, our answer to this problem is 1 4/15.

Subtracting Mixed Numbers with Borrowing

Another problem you may run into while subtracting mixed numbers is that your fractions
cannot be subtracted because the first fraction is smaller than the second fraction.
(For example, 1/8 – 3/8 cannot be subtracted). In this case, you need to borrow
from the whole number. This is a little tricky, so follow these steps to be sure
and get the right answer! Here’s a sample problem that you can follow along with:

Notice that if we tried to subtract the fractions, we would get 1/8 – 3/8 and would
not be able to complete the problem. Therefore, we’re going to go back to the first
number and borrow from the 3. You borrow “one” just like if you were borrowing in
a normal subtraction problem. Cross out the three, and change it to a 2. Now, we
have to add “1” to 1/8, which would look like this:

This doesn’t look very helpful, so we need to change the 1 into its fraction form.
We know that 1 as a fraction is simply any number over itself. For this example,
since my denominator is 8, I’m going to use 8, like this:

Now, I have to add together the original fraction and “1”, which would look like
this:

Now, I have an improper fraction, which is exactly what I want! Anytime you borrow
to solve a subtraction problem, you should have an improper fraction at this point.
Now, I can continue with my fraction subtraction, which looks like this:

Now, continue the whole number subtraction with the numbers you have, 2 – 1 = 1
(note: we’re using 2, not 3, because we borrowed from the 3 and made it a 2).

My answer looks like this now:

The last step is to check the fraction and make sure it is in its simplest form,
reduced. In this case, I need to reduce the fraction. That looks like this:

So, my FINAL answer is:

One more type of problem you may encounter is subtracting mixed numbers with uncommon
denominators. This is essentially the same as subtracting with common denominators;
you just have to remember to find a common denominator before you borrow. Here’s
an example to help you see what this would look like.

First, you would find common denominators for the fractions. We know that the least
common multiple between 8 and 4 is 8, so we would use 8 as our common denominator.
We have to expand our second fraction so that it has a denominator of 8. That looks
like this:

Now we can continue the subtraction like we did in the first example. Our new fraction
subtraction would look like this:

We can’t do this subtraction because our second fraction is larger than our first
fraction, therefore we have to borrow from our first whole number (3) in order to
complete this problem. When we borrow from the 3, we cross it out and turn it into
a 2. Then we add “1” to 1/8. That looks like this:

Then, we have to re-write 1 in its fraction form, which looks like this:

Last, we add together the two fractions, like this:

So now, our new fraction subtraction, after the borrowing, looks like this:

Then, go back and subtract the whole numbers, so 2 – 1 = 1 (we’re using 2, because
we borrowed 1 from the 3 and changed it into a 2).

Thus, our answer is 1 7/8.

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