$Rules and Definitions$

Detailed Help For Adding Fractions With Different Denominators

You have already seen how easy adding fractions with the same or like denominators can be. You simply add the numerators and keep the same denominator, then simplify if needed. Now we are going to talk about adding fractions with different denominators.

When you finish this lesson, you will wonder why you ever worried about adding these fractions in the first place. Promise!

First of all, when adding fractions with different denominators, the first step says that we must change these fractions so that they have the "same denominator".

Here are the steps for adding fractions with different denominators. We will break-down each step just like before to make sure you've got it. Then we will add some tougher numbers. And finally, we will help you pull everything together. Okay!

So, here are the steps.

Build each fraction so that both denominators are equal.

Remember, when adding fractions, the denominators must be equal. So we must complete this step first. What this really means is that you must find what is called a Common Denominator. For our advanced lesson you will be required to work the problem using what's called the Least Common Denominator (LCD). In either case you will build each fraction into an equivalent fraction.
Re-write each equivalent fraction using this new denominator
Now you can add the numerators, and keep the denominator of the equivalent fractions.
Re-write your answer as a simplified or reduced fraction, if needed.

We know this sounds like a lot of work, and it is, but once you understand thoroughly how to find the Common Denominator or the LCD, and build equivalent fractions, everything else will start to fall into place. So, let's take our time to do it Right!

But keep in mind, if you are doing homework, be sure to answer the problems in the form asked for in the assignment.

Okay let's start with...

The Basics

Add 1/2 + 1/3

1/2

1/3

Notice that the overall size of our point of reference
(The Whole) is EXACTLY the same.

Step #1 in our rule tells us that the denominators must be equal. And the easiest way to find a common denominator is to just multiply the denominators.

So let's do that now...

2 x 3 = 6

The Common Denominator for 1/2 and 1/3 is 6

Step #2 - Re-write each equivalent fraction using this new denominator.

Since...

1/2

1/2 is equivalent to 3/6

1/6

And...

1/3

1/3 is equivalent to 2/6

1/6

We re-write our equation to read...

Add: 3/6 + 2/6

Now we are ready to do Step #3 - ADD the numerators, and keep the denominator of the equivalent fractions (which is 6).

So, we end up with...

3/6 + 2/6 = (3 + 2 )/6 = 5/6

3/6

2/6

5/6

Finally, Step #4 - Re-write your answer as a simplified or reduced fraction, if needed.

In our example, the answer (5/6) is already in its simplest form. So, no further action is required!

That's It!

A quick and easy way to add fractions with different denominators.

Now it's time to take the Advanced Course for adding fractions with different denominators.

Are you READY?

I'm going to take you step-by-step through the whole process. These examples give you what you'll need to work the harder problems in adding fractions. So be prepared!

And it all starts with...

Finding A
Common Denominator

Sometimes, you may not be concerned with what number is in the denominator, as long as you can come up with the correct answer. Well, the "quick 'n dirty" formula for adding two fractions with different denominators looks like this...

Actually, it's ALGEBRA!

YIKES!

Didn't know you needed algebra for adding fractions?

Don't Worry!

This formula is for those OLDER folks... to make it a little easier for them to get the idea quickly. And if YOU "get it", Adding fractions with different denominators will be a snap for you.

But for the rest of us... we are going to take a simpler approach...

You should know that the fastest way to find a common denominator is to multiply the denominators.

That's it!

If you wanted to add 1/3 + 1/4, to find the common denominator you just multiply 3 x 4. So, the common denominator is 12. Even if you were adding three fractions like 1/8 + 4/7 + 3/56, the same rule applies. Multiply 8 x 7 x 56 to get a common denominator of 3136.

You may have already figured out that this might not work out to well for a lot of the fractions you'll be working with. Here's the problem. As the denominators get bigger, it gets harder to work with them. And sometimes, this can be a real problem.

How about adding 27/28 + 1/56 + 13/35. Well, here's your common denominator.

It's... 54880

WOW! That's a really BIG number, and it's only the FIRST STEP.

So there has to be a better way for adding fractions with different denominators.

And there is, if you know how to...

Find the Least Common
Denominator (LCD)

The least common denominator of two or more non-zero denominators is actually the smallest whole number that is divisible by each of the denominators. There are two widely used methods for finding the least common denominator.

Actually, this is the same basic idea behind finding the Least Common Multiple (LCM) for whole numbers (without the fractional parts).

Note: In the examples below, we'll be adding three fractions instead of the usual two because the principles are the same. This will give you a better understanding of the process. And in the "Pulling Everything Together" section, we will be adding four fractions.

Let's take a look at...

Method 1:

Simply list the multiples of each denominator (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.

Example: Suppose we wanted to add 1/5 + 1/6 + 1/15. We would find the least common denominator as follows...

First we list the multiples of each denominator.

Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,...
Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,...
Multiples of 15 are 30, 45, 60, 75, 90,....

Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.

Therefore, the least common denominator of 1/5, 1/6 and 1/15 is 30.

This method works pretty good. But, just like we noted above, adding fractions with larger numbers in the denominators it can get pretty messy.

So hold that thought for a moment, as we look at another way to find a least common denominator for adding these same fractions.

Method 2:

Factor each of the denominators into primes. Then for each different prime number in all of the factorizations, do the following...

Count the number of times each prime number appears in each of the factorizations.
For each prime number, take the largest of these counts.
Write down that prime number as many times as you counted for it in step #2.
The least common denominator is the product of all the prime numbers written down.

Example: We'll use the same fractions as above: 1/5, 1/6 and 1/15.

Factor into primes (Click here to see our table of prime numbers.)
- Prime factorization of 5 is 5 (5 is a prime number)
- Prime factorization of 6 is 2 x 3
- Prime factorization of 15 is 3 x 5

Notice that the different primes are 2, 3 and 5.

Now, we do Step #1 - Count the number of times each prime number appears in each of the factorizations...
- The count of primes in 5 is one 5
- The count of primes in 6 is one 2 and one 3
- The count of primes in 15 is one 3 and one 5
Step #2 - For each prime number, take the largest of these counts. So we have...
- The largest count of 2s is one
- The largest count of 3s is one
- The largest count of 5s is one
Step #3 - Since we now know the count of each prime number, you simply - write down that prime number as many times as you counted for it in step #2.

Here are the numbers...

2, 3, 5
Step #4 - The least common denominator is the product of all the prime numbers written down.

2 x 3 x 5 = 30
Therefore, the least common denominator of 1/5, 1/6 and 1/15 is 30.

As you can see, both methods end up with the same results.

The reason we might want to use the different methods is because Method #1 works great for small numbers. But when the numbers get bigger, Method #2 is the ONLY way to go.

Now let's make the tricky part, really easy -- convert each fraction to an equivalent fraction using the newly found LCD, which is 30.

Remember our problem: Add: 1/5 +1/6 + 1/15

Step #2 for adding fractions with different denominators says - "Re-write each equivalent fraction using the LCD as the denominator." So let's do it!

This is going to get a little detailed, so hang in there!

Re-write Each Fraction As
An Equivalent Fraction

The Rule to re-write a fraction as an equivalent fraction using the LCD says...

Divide the LCD by the denominator of the fraction.
Multiple the answer times the numerator of the fraction.
Re-write the fraction using the LCD as the denominator.

So, if we write 1/5 as an equivalent fraction using 30 as our denominator, we have 30 divided by the denominator "5", which equals 6. We then multiple that 6 times the numerator "1" which gives us the new numerator of 6.

Finally, we re-write the equivalent fraction using the 30 as our denominator, therefore our equivalent fraction is 6/30.

The Rule actually looks like this...

New Numerator = (LCD � Denominator) x Numerator
New Denominator = LCD

Now we repeat the process for 1/6 and 1/15

Using 1/6 next, (30 � 6 = 5, and 5 x 1 = 5), so 1/6 is equivalent to 5/30

And for 1/15, (30 � 15 = 2, and 2 x 1=2), so 1/15 is equivalent to 2/30

Now then, at long last we can add our fractions...

1/5 + 1/6 + 1/15 = 6/30 + 5/30 + 2/30 = 13/30

Pulling Everything Together

We are going to use a little tougher problem for adding fractions with different denominators to illustrate that you CAN do it. Also, we will use Method #2 to find the LCD because it works best in almost every case.

If you have problems with any parts of this exercise, re-read the section above that covers it. We are going to talk through each step for adding these fractions without citing the rules exactly, just like in the "real world".

Look at how everything works and you will be just fine!

We'll be adding these fractions... 1/9 + 1/8 + 5/12 + 7/18

Since our denominators are 9, 8, 12, and 18, we need to find the LCD. So we factor each number into primes.

Factorization of 9 is 3 x 3
Factorization of 8 is 2 x 2 x 2
Factorization of 12 is 2 x 2 x 3
Factorization of 18 is 2 x 3 x 3

When we do our largest count of the prime numbers, we find three 2s, and two 3s (do you see them?), so we re-write the count and find the product. Like this...

2 x 2 x 2 x 3 x 3 = 72

Now we have our least common denominator of 72

Next... we convert each fraction to an equivalent fraction using 72 as our new denominator. So, let's convert...1/9 + 1/8 + 5/12 + 7/18

1/9 = 8/72
1/8 = 9/72
5/12 = 30/72
7/18 = 28/72

Mission accomplished... all of our denominators are the same, so we can just add up the numerators.

Our new equivalent fractions are 8/72, 9/72, 30/72 and 28/72

That's what changes our problem to 8/72 + 9/72 + 30/72 + 28/72

Now adding all of the numerators and placing the results over our common denominator, the answer is...

8/72 + 9/72 + 30/72 + 28/72 = 75/72

Since our answer is an improper fraction (the numerator is larger than the denominator), we now have three options.

Show the answer as is, with the least common denominator.
75/72
Reduce the fraction and show it as the lowest reduced equivalent.
25/24
Simplify the answer, and show it as a mixed number.
1 1/24

Remember, always show your answer in the form asked for in your instructions.

The following information is a repeat of information found in other lessons, just in case you missed them.

How To
Simplify Your Answers

Sometimes when adding fractions of any type, you will need to simplify your answer. What that really means is that you must show your results in the "best" form possible. As a result, here are a few more things to think about...

First, your answer may be a higher equivalent fraction, which is better represented in its reduced form. Many teachers will insist that you reduce your answer, whenever possible.
Also, adding fractions will often result in what's called an improper fraction. This is where the numerator is larger than the denominator. To write these answer in their simplest form you will have to convert them to a mixed number. This will show a representation of the Whole Parts and the Fractional Parts.
And finally, you may also be adding mixed numbers where the fractional parts have a different denominator. Even with these types of problems, you may need to simplify your answers.

So let's continue with some detailed information about these special cases.

Reducing Fractions To
Their Lowest Equivalent

Here's the situation. You have added the fractions okay, but your answer may not be showing the lowest equivalent fraction. So how do you make sure your answer is shown in its lowest equivalent?

Let's use an easy example so you will get the idea...

Notice that the original answer after adding the fractions is "2/4." To determine if our answer is in its simplest form, we must factor the numerator and the denominator into its prime numbers. Click here for a review of prime numbers.

What we are looking for are the prime numbers that are common to both the numerator and the denominator. If we find these common numbers, we can then cancel them out. The results will be the lowest equivalent fraction.

Since "2" is a common factor in both the numerator and denominator of our example, it indicates that our answer is not in its simplest form. Therefore, we will cancel out (/) one of the 2's in both the numerator and denominator by dividing by "2". The results is a reduced fraction in its simplest form.

Here's the Rule...

Always keep in mind...

Whatever you do to the numerator of a fraction you must also do to the denominator. So if you have to divide the numerator by a number, you must also divide the denominator by the same number. That way you will not change the overall value of the fraction.

Let's do a little tougher problem to be sure you've got it...

In this problem, a "2" and a "3" can be found in both the numerator and the denominator. Notice how we only cancel-out one-for-one! First we divide the numerator and denominator by "2", then divide both the numerator and denominator by "3." So what is left in the numerator is 1 x 1 x 3 = 3, and the denominator is 1 x 2 x 2 x 1 = 4. That leaves use with a reduced fraction equal to 3/4.

Got it?

GREAT!

Now let's look at...

How to Simplify
Improper Fractions

You may remember that an improper fractions is where the numerator has a greater value than that of the denominator. So each time you add two fractions and your answer ends up as an improper fraction, you must simplify your answer. The results will be in the form of a mixed number.

To convert an improper fraction into a mixed number, just divide the numerator by the denominator. The results will be a whole number part and a fractional part.

Here is an example...

As you can see, this is a pretty straightforward operation. But keep in mind that if there is no remainder, the answer is the WHOLE NUMBER only.

Now that you are the master of adding fractions with the same denominator, it is time to tackle a tougher problem...

Adding Mixed Numbers
With Different Denominator

The easiest way to work with mixed numbers is to convert them to improper fractions first, then convert your answer back to a mixed number.

But first...

Here's the Rule for converting mixed numbers into improper fractions...

To actually do a conversion, it would look like this...

Putting this problem into words...

... to convert 2 1/8 to an improper fraction, we just multiply the Whole Number (2) times the denominator (8), and add that answer to the Numerator (1). The result is the improper fraction 17/8.

Now let's put this new found knowledge to work and add a couple of mixed numbers.

Note: We are using fractions with the same denominator here simply to point out that you must convert the mixed number into an improper fraction first.