Common Denominators

There are three different ways to find a common denominator. One of them can only be used in certain cases, while the other two can be use with any two fractions. Each one has advantages and disadvantages.

However, there are important reasons for students to learn the most general strategy first. That’s the strategy taught in the lesson in the video above. The three strategies are:

• Using the product of the denominators as the common denominator
• Using the Least Common Multiple as the common denominator
• When one denominator is a multiple of the other, using the larger denominator as the common denominator

In this course, we start with the first approach to finding a common denominator listed above – using the product of the denominators as the common denominator - for several reasons.

The most important reason is that mathematically, it is the most general strategy, and can be used with all fractions.

The approach applies to finding common denominators with fractions having only numbers in the numerator and denominator, and can be used with any two such fractions. In addition, however, the same strategy can be used later in Algebra to find common denominators for adding and subtracting fractions with variables in them. These fractions that contain variables are called “rational expressions,” but you need not be concerned with this situation until you reach Algebra.

Learning the method taught in this lesson, however, will mean that it will be much easier to understand the rationale behind the method that is required in later in Algebra for rational expressions. This general strategy is the only strategy that can be used for adding and subtracting fractions (rational expressions) once variables are introduced. (It should also be noted that the method taught in the lesson above is the method specified in the Common Core State Standards for math.)

A second strategy that is taught in this course is based on using the Least Common Multiple to find a common denominator. It is also a general strategy in that it can be used for any two fractions written with numbers only. However, it cannot be used as a general strategy when variables are included in the fractions in Algebra.

For that reason, the most general strategy that can be applied to finding a common denominator for any two fractions is the strategy taught in the lesson in the video above. That strategy is based on multiplying the denominators of the two separate fractions to find the common denominator.

Another important reason for learning this general strategy is that it makes it easy to compare the size of any two fractions using a common denominator to determine which is larger or smaller.

With an understanding of the product of the denominators as a common denominator, students can learn a quick an easy method of comparing fractions based on “cross-multiplication,” without the need to calculate the product as the denominator. Because students learn the rationale for the approach, and understand why the cross-multiplication works, the process is not learned as a “rote process” without meaning. Instead, it becomes a meaningful process that also has the advantage of being quick and easy. An example is given later below.

The only downside of using this general strategy to find a common denominator is that the result is not always the “lowest common denominator,” or “least common denominator.” That means that the fraction that results after using this approach to finding a common denominator, then adding or subtracting fractions, is not always in simplest terms, or lowest terms.

If you are required to express the result of adding or subtracting the fractions in lowest terms, you may have to simplify the fraction that you get to lowest terms, which is often referred to as “reducing” the fraction. The process of simplifying fractions and reducing fractions to lowest terms are taught in other lessons which you can access when you sign up for all free lessons.

Here’s an example using this general strategy for adding two fractions:

1/4 + 5/6 = __.

In this example, the common denominator is the product of the two denominators – 4 and 6 – so the common denominator is 24. When the fractions are written with this denominator, the problem becomes:

6/24 + 20/24 = 26/24

The sum is 26/24. This fraction could also be rewritten in simplest terms as 13/12.

A second method that can be used for all fractions that have only numbers in them is to used the Least Common Multiple as the common denominator. That method is covered in a different lesson.

To illustrate the differences in this approach, we can add the same two fractions in the last example using the Least Common Multiple:

1/4 + 5/6

The first step would be to find the Least Common Multiple for 4 and 6 as follows:

Some multiples of 4 include:

4 8 12 16 20

Some multiples of 6 include:

6 12 18 24 30

The smallest multiple that is common to both is 12, so 12 is the Least Common Multiple, which would become the common denominator. The problem would then become:

3/12 + 10/12 = 13/12

The sum of these is 13/12, so you get the same result as before.

When comparing the steps needed to complete each of these, it is simpler to multiply the denominators to get a common denominator than to find the Least Common Multiple. (However, there may be an additional step required in the more general strategy to write the result in lowest terms, as in the example above.)

Here’s an example of two fractions that can be added using a third strategy.

1/4 + 5/12 = __.

In this case, 12 is a multiple of 4, so the easiest way to get a common denominator is to use the larger denominator, 12. That means only the first fraction must be rewritten, which gives this:

3/12 + 5/12 = 8/12

Note that this result would then need to be rewritten in lowest terms, 2/3, if that is required.

In all cases in which the denominator of one fraction is a multiple of the other denominator, the simplest strategy is to use the larger denominator as the common denominator. There is nothing to be gained by using either the product of the denominators or the Least Common Multiple instead.

Finally, the use of the most general strategy – the product of the denominators - makes it easy to compare fractions using a cross-multiplication strategy. Here’s an example.

Which fraction is larger?

8/7 or 6/5

To quickly determine the answer, it is only necessary to multiply 7 times 6 and 5 times 8 and compare the result. This gives the numerator for the two fractions that would result by rewriting them with a common denominator of 35, the product of the two denominators.

In this case, the numerator of the first fraction would become 40, which is the result of cross-multiplying the denominator of the second fraction, 5, times the numerator of the first fraction, 8.

The numerator of the second fraction would become 42, which is the result of cross-multiplying the denominator of the first fraction, 7, times the numerator of the second fraction, 6.

That means the second fraction, 6/5 is the larger fraction. We know that because both resulting fractions would have the same denominator, 35, which is the product of the two original denominators.

If written with a common denominator, the two fractions would become:

40/35 or 42/35

Since 42 is greater than 40, the second fraction is the larger of the two.

In the lesson shown in the video above, the use of the product of the denominators as the common denominator for adding or subtracting fractions is illustrated using number lines. There are two reasons for this approach.

First, before teaching the strategy for finding a common denominator, the lesson first illustrates why a common denominator is needed. This is something that is rarely taught, but makes the process of finding a common denominator much more understandable and intuitive.

Second, the use of the number lines makes it easy to see why this approach will always result in a common denominator. This means that students are able to understand why this approach is taught, as well as how to use it.

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