Comparing Fractions

In this lesson, you will learn about comparing fractions. Before you start this lesson, I recommend that you study or review my lesson about fractions. You could also take the quiz below to see how well you can compare two fractions.

Before I show you two ways to compare fractions, you will need to understand the meaning of cross product, the inequality sign, and common denominator.

Cross product:

The cross product is the answer obtained by multiplying the numerator of one fraction by the denominator of another. For instance, how do we  get the cross products of the two fractions below?

We can do 2 × 6 = 12 and 3 × 5 = 15.

Meaning of inequality sign:

The sign (>) means greater or bigger than.

For instance, 6 > 4

The sign (<) means smaller than

For instance, 4 < 6

Common denominator:

When two or more fractions have the same denominator, we say that the fractions have a common denominator. The following three fractions all have a common denominator.

Now you can compare fractions to see which one is bigger either by using the cross product or by looking for a common denominator.

Comparing Fractions using the Cross Product

Do a cross product to see which of the two fractions below is bigger.

Start your cross product by multiplying the numerator of the fraction on the left by the denominator of the fraction on the right. You get 2 × 4 = 8

Then, multiply the numerator of the fraction on the right by the denominator of the fraction on the left. You get 3 × 3 = 9

Put 8 beneath 2/3 and 9 beneath 3/4

2 3	3 4
8	9

Since 8 is smaller than 9, 2/3 is smaller than 3/4

Do a cross product to see which of the two fractions below is bigger.

5 × 8 = 40 and 6 × 6 = 36

Put 40 beneath 5/6 and 36 beneath 6/8

5 6	6 8
40	36

Since 40 is bigger than 3, 5/6 is bigger than 6/8

A second method to use when comparing fractions is to first get a common denominator

 Let us compare again the following two fractions using common denominator.

Notice that you can multiply the denominator for the first fraction, which is 6 by 8 and multiply the denominator for the second fraction, which is 8 by 6 to get your common denominator.

We get the following two equivalent fractions.

Since 40 is bigger than 36, 5/6 is bigger than 6/8

The following rules are helpful!

Rule #1

When two fractions have the same denominator, the bigger fraction is the one with the bigger numerator. Does that make sense?

Let's again use our Pizza in the lesson about fractions as an example. If your pizza has 10 slices and you eat 5. This situation is represented by the fraction below.

If you eat one more, that is 6 slices

Now, is it clear and obvious that 6/10 is bigger than 5/10?

Rule # 2

When comparing fractions that have the same numerator, the bigger fraction is the one with the smaller denominator.

Once again, let us use our pizza as an example. Say that you bought two large pizzas and they are the same size.

Let's say that the first pizza was cut into 10 slices and the second was cut into 15 slices. No doubt if the second pizza is cut into 15 slices, slices will be smaller.

If you grab 2 slices from the first, the expression for the fraction

If you grab 2 slices from the second, the expression for the fraction

Slices for the latter will definitely be smaller. Therefore, 2/10 is smaller than 2/15.