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Robert Ferdinand has taught university-level mathematics, statistics and computer science from freshmen to senior level. Robert has a PhD in Applied Mathematics.

When solving rational equations, just like with any other fractions, you must first find the least common denominator. This lesson will show you how to find the least common denominator and also solve the rational equations.

Least Common Denominator

Have you ever read a sentence that seemed to be in English, but most of the words just made no sense? Math can be that way sometimes too. That is why it is important to have a good grasp on what the terms mean. That way, when you read a mathematical sentence, you can easily decipher what its meaning is.

This is a very real concept when dealing with the idea of a least common denominator, or LCD, of two or more fractions. The mathematical definition of lowest common denominator reads, 'the smallest positive integer that is a multiple of each denominator in the set.' You might have just read that and thought, 'Is she speaking a foreign language?' But it is all English and can be easily understood if we first define all of the math terms:

An integer is a number that can be written without a fraction or a decimal. 4, 18, and 2,305 are all examples of integers.

A multiple is a product of two or more numbers.

And product just means the answer to a multiplication problem.

The denominator is the bottom number of a fraction.

And finally, a set is a group of two or more numbers, in this case, fractions.

So you can see that we are looking for a number that is not a fraction itself but that is the product of each bottom number of the fractions we are looking at and another number.

How to Find the Least Common Denominator

There are many different ways to find the least common denominator. I will show you two, and then you can pick which one works best for you. Both methods will give you the same number every time.

The first method is just to list the factors of each number until you find a matching one. Let's use 1/21 and 1/6 as our examples. Find the least common denominator of these two fractions. First, we identify the denominators as 21 and 6. Next, we list out all the factors of 21. The way to do this is, starting with 1, multiply each number by 21.

1 * 21 = 21

2 * 21 = 42

3 * 21 = 63

4 * 21 = 84

5 * 21 = 105

You can keep this pattern going for a long time; however, I usually stop with five. If that isn't enough, it will become apparent and we can continue. Then we do the same thing with the other denominator. We are looking for a matching factor.

1 * 6 = 6

2 * 6 = 12

3 * 6 = 18

4 * 6 = 24

5 * 6 = 30

6 * 6 = 36

7 * 6 = 42

We can stop there, because 42 shows up in both lists. This means that it is the least common denominator. This method can be long and cumbersome, but it is relatively easy because there's not much to it. The second method involves using the following formula.

The greatest common divisor is the biggest number that will divide into each number evenly. For 21 and 6, the greatest common divisor is 3. So, to fill in the formula, we get the least common denominator of 21 and 6 is equal to 21 / 3 * 6, which is equal to 7 * 6, or 42.

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This method will not take as much time, but you will have to determine the greatest common divisor. Let's try one more example using both methods. Find the least common denominator of 2/3 and 1/12. Using the first method, we can create this chart.

Common denominators of 3 and 12

The second method gives us the least common denominator between 3 and 12 is equal to 3 / 3 * 12, which is equal to 1 * 12, or 12. Each method gives the same answer, so feel free to use whichever method works for you.

Rational Equations

A rational equation is an equation in which both sides contain fractions with polynomials. Here is an example.

How to Solve Rational Equations

The purpose for solving a rational equation is to determine the value for x. The first thing that needs to happen is that the fractions need to have common denominators. Look at this first example.

Because these fractions have the same denominator, it is easy to see what x is equal to. x = 2. That is why making sure your fractions have the same denominator makes solving rational equations so easy. If they are equal, then all you have to do is make the numerators equal each other and solve for x.

Let's try another example.

x

Here is one more example using our rational equation from the example above.

xxx

The final step is to plug the answer into each original denominator to make sure that we don't have a 0. This would cause the solution to be undefined, since no fraction can have a 0 in the denominator.

x - 2 is equal to (-2) - 2, which is -4. So, x is equal to -2. Since neither of these solutions is 0, x can equal -2 and that is our answer.

Lesson Summary

The process for solving rational equations might seem like a foreign language but really just involves a few simple steps as long as you know what all the words mean. The first step is to find the least common denominator, then write your expressions so each term has a common denominator. Finally, solve the numerator portion of the equation for x.

Learning Outcomes

By the end of this lesson you should be able to find the lowest common denominator in order to solve rational equations.

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