Previously I mentioned that one of the cardinal rules of algebraic fractions was that we would always fully factor the numerators and denominators, and then we would reduce by canceling factors that appeared in both the numerator and denominator. I then mentioned that there was one exception to that rule, and it's in this section that we'll see that exception. Here we're going to explore the rules for adding fractions. In a nutshell, the process is:

Example #1

Add +

Solution #1

Remembering our rule to always factor, and then reduce, we write this as:

+ = +

Now that we've factored and reduced each fraction, we ask ourselves "What is the least common multiple of those two denominators?" For this simple problem, you probably can immediately recognize that the answer is 6. We'll go with that for now, but a little later, I'll give you an example with more "messy" denominators, and we'll talk about a more precise method for finding the LCD.

Now that we've found the LCD = 6, we're going to determine what we have to multiply each denominator by to get 6. For example, since the first fraction has a denominator of 3, we'd have to multiply by 2 to get 6. But, since we're not allowed to change the value of the fraction, only its appearance, we have to multiply both the numerator and the denominator by the same value.

=  · =

Similarly, =  · =

So  + = +

Now here is the one place where you do NOT want to factor and reduce. In the step where you convert the fractions to the same denominator, it will always be possible to factor and reduce - you can factor out and cancel the thing you just multiplied by. But doing that just gets you right back where you started. You don't want to do this!

I have a little trick that may help you remember not to cancel at this point. Instead of writing it this way:

+

Write it this way:

+

Of course, this would be dangerous to do if you had any variables that were capital L, C, or D, but if you don't have those, putting LCD there will prevent you from doing any canceling. Then, when you get to the end of the problem, you've just got to remember to write out the actual LCD instead of the acronym. The other advantage to doing this is, if the LCD is something horrific like (x - 2)(x + 3)(x - 1)², LCD is a lot less writing!

Our final rule for adding is, we add the numerators, while keeping the denominator the same:

+ = =

Example #2

Add +

Solution #2

As always, we begin by factoring and doing any canceling (in this case, we can factor, but there's no cancelation we can do).

+ = +

In this case it's not quite as obvious what the LCD is. However, since your denominators are written in factored form, there's a really nice rule you can use to find the LCD:

In this case, the first fraction's denominator contains a 2 and 3, and the second denominator also contains a 2 and a 3. Thus, our LCD is going to consist of twos and threes multiplied together. How many of each? Well, the highest exponent for 2 is 4 (in the second denominator) and the highest exponent for 3 is 2 (in the first denominator). Thus, the LCD is 2⁴·3².

Furthermore, since we have our denominators and our LCD written in factored form, it is very easy to spot what is missing from each denominator:

 = · =

=  · = 

  + = =

Remember our rule to factor and reduce each fraction; we should factor that 119, to see if anything can cancel. 119 = 7·17, and since the denominator does not contain either of those factors, the fraction is irreducible.

Example #3

Add +

Solution #3

You know the drill. Rewrite with parentheses around the binomials to remind yourself that the fraction bar is a grouping symbol. Factor what you can in each fraction, and then reduce:

  + = + = +

Our denominators are fully factored, so the LCD has to contain both (x - 1) and (x + 2). Since neither of them has exponents, the LCD is simply: (x - 1)(x + 2).

=  · = =

=  · = =

Therefore, + = =

Note that in the final step, all we did was combine like terms. Then, since the numerator is not factorable, this fraction is in simplest form.

The Rules So Far

1. Any time you see a fraction, FACTOR the numerator and denominator.
2. Any time you have a fraction with factored numerator and denominator, REDUCE, by canceling common factors.
3. Multiplying fractions involves multiplying across the numerators, and multiplying across the denominators.
4. Dividing fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.
5. To add fractions, perform the following steps:
   1. factor and reduce each fraction (steps 1 and 2)
   2. find the LCD
   3. convert each fraction to the LCD (and skip steps 1 and 2 on the new fractions!)
   4. add the numerators, using the distributive property if necessary, and combining any like terms you find

As always, once you reach that last step, fully factor the result to find out if anything will cancel (repeat steps 1 and 2).

1.

Add +

2.

Add +

3.

Add +

4.

Add +

5.

Add +

6.

Add +

7.

Add +

8.

Add +

9.

Add +

10.

+