Solving Rational Equations and Finding the Least Common Denominator

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  • 0:06 Least Common Denominator
  • 1:24 Finding the Least…
  • 4:03 Rational Equations
  • 4:11 Solving Rational Equations
  • 6:20 Lesson Summary
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Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Expert Contributor
Robert Ferdinand

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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Practice Problems (please show all of your work):

Solve the following rational equations by using the least common denominator (LCD):

(a) 1 / x + 1 / (2x) = 3.

(b) 3 / x = 10 / (x + 2).

Answers (to check your work):

(a) To solve this equation, we'll use the following steps.

Step 1) Find the least common denominator, which is LCD = 2x.

Step 2) Multiply each term of the equation by the LCD to get:

  • (2x)(1 / x) + (2x)(1 / (2x)) = (2x)(3)
  • 2 + 1 = 6x
  • 3 = 6x
  • (3 / 6) = (6x / 6)
  • x = 1/2

Step 3) Check to make sure that the solution from Step 2 does not make the denominator zero in any of the terms of the original equation.

This checks out; hence, x = 1/2 is the solution of our equation.

(b) Again we'll use the following steps:

Step 1) Find the least common denominator, which is LCD = x(x + 2).

Step 2) Multiply each term of the equation by the LCD to get:

  • x(x + 2)(3 / x) = x(x + 2) (10 / (x + 2))
  • 3(x + 2) = 10x
  • 3x + 6 = 10x
  • 3x + 6 - 3x = 10x - 3x
  • 6 = 7x
  • x = 6/7

Step 3) Check to make sure that the solution from Step 2 does not make the denominator zero in any of the terms of the original equation.

This checks out; hence, x = 6/7 is the solution of our equation.

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