$Rules and Definitions$

Subtracting Fractions Is Actually Easier Than Adding Fractions

When subtracting fractions with different denominators, we follow the same process we used for adding different fractions. But since everybody doesn't start with addition, we provide the same level of detail for subtraction.

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

Here are the steps for subtracting fractions with different denominators. We will break-down each step just like before to make sure you've got it. Then we will subtract 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 subtracting 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. Most of the time 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 subtract the numerators, and keep the denominator of the equivalent fractions.
Re-write your answer as a simplified or reduced fraction, if needed.

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

Subtract: 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...

Subtract: 3/6 - 2/6

Now we are ready to do Step #3 - Subtract 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 = 1/6

3/6

2/6

1/6

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

In our example, the answer (1/6) is already in its simplest form. So, no further action is required!

That's It!

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

Just like with addition, there is an Advanced Course for subtracting fractions with different denominators. But, it's not nearly as intense.

So are you READY to go?

As usual, 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 subtracting fractions.

Actually, there really aren't any harder problems. Just a couple of Rules to keep things straight.

Once again, for those that missed it, well repeat 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. Here is the "quick 'n dirty" formula for subtracting two fractions with different denominators looks like this...

This formula is just an algebraic expression that show how all of the numbers fit together. So if you're up to the challenge, just plug in your fractional numbers and do the math. It's really not that tough.

If you look at the fraction to the right of the equal sign, you'll notice that the denominator (b x d) tells you to simply multiply the denominators, So, to find a common denominator, that's ALL you would do.

Period.

That's it!

So, If you wanted to subtract 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 subtracting three fractions like 7/8 - 4/7 - 3/56, the same rule applies. Multiply 8 x 7 x 56 to get a common denominator of 3136.

But, you're faced with a problem.

As the denominators get bigger, it gets harder to work with them. And sometimes, this can be a real problem.

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

It's... 54880

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

But sure enough, there is a better way to subtract fractions with different denominators.

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).

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 subtract 1/5 - 1/6. 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,...

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 and 1/6 is 30.

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

So let's look at another way to find the least common denominator to subtract 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, and 1/6.

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

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
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 they are...

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: Subtract: 1/5 -1/6

Step #2 for subtracting 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 time 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

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

Now we can subtract our fractions...

1/5 - 1/6 = 6/30 - 5/30 = 1/30

Pulling Everything Together

To prove that YOU can do it, we'll throw in a couple of fractions with bigger numbers. And then, go through the same step as before but without reciting the rules exactly. Just to make double-sure you've got it.

We'll be subtracting these fractions... 17/18 - 4/9 - 1/8

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

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

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... 17/18, 4/9, and 1/8

17/18 = 68/72
4/9 = 32/72
1/8 = 9/72

Okay! All of our denominators are the same, so we can just find the difference of numerators.

Our new equivalent fractions are 68/72, 32/72, and 9/72

And that changes our problem to 68/72 - 32/72 9/72

Finally, we do the subtraction of the numerators and place the results over our common denominator, the answer is...

68/72 - 32/72 - 9/72 = 27/72

Since our answer is not in its simplest form, we have two options.

Show the answer as is, with the least common denominator.
27/72
Reduce the fraction and show it as the lowest reduced equivalent.
3/8

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.

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 subtract 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 subtracting fractions with the same denominator, it is time to tackle a tougher problem...

Subtracting 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 subtract a couple of mixed numbers.

Suppose we wanted to subtract 4 1/2 - 3 1/3.

First we would convert our mixed numbers using the Rule from above...