# 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.

Before I show you two ways to compare fractions, you will need to learn the following.

Cross product: The answer obtained by multiplying the numerator of one fraction by the denominator of another

For instance, to get the cross products of the fractions below:

 2 / 5 and   3 / 6

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.

 2 / 6 ,  1 / 6 and   5 / 6  all have a common denominator

Now, here are two ways to compare fractions.

The first way is to do a cross product.

 Let us compare   2 / 3 and      3 / 4

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, then   2 / 3 <      3 / 4

 Let us compare   5 / 6 and      6 / 8

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 36, then   5 / 6 >      6 / 8

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

 Let us compare again   5 / 6 and      6 / 8

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.

Warning! Whatever you multiply the denominator, you have to multiply your numerator by the same thing so that you are in fact getting equivalent fractions.

 5 / 6 becomes     40 / 48

 6 / 8 becomes     36 / 48

 Since 40 is bigger than 36, then   40 / 48 >     36 / 48

 Therefore,   5 / 6 >    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.

That is
5 / 10

If you eat one more, that is 6 slices

That is
6 / 10

Now it has become obvious that

 6 / 10 >   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
2 / 10

If you grab 2 slices from the second, the expression for the fraction
2 / 15

Slices for the latter will definitely smaller.

 Therefore,   2 / 10 >    2 / 15

This lesson about comparing fractions is over.

Before I show you two ways to compare fractions, you will need to learn the following.

Cross product: The answer obtained by multiplying the numerator of one fraction by the denominator of another

For instance, to get the cross products of the fractions below:

 2 / 5 and   3 / 6

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.

 2 / 6 ,  1 / 6 and   5 / 6  all have a common denominator

Now, here are two ways to compare fractions.

The first way is to do a cross product.

 Let us compare   2 / 3 and      3 / 4

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, then   2 / 3 <      3 / 4

 Let us compare   5 / 6 and      6 / 8

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 36, then   5 / 6 >      6 / 8

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

 Let us compare again   5 / 6 and      6 / 8

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.

Warning! Whatever you multiply the denominator, you have to multiply your numerator by the same thing so that you are in fact getting equivalent fractions.

 5 / 6 becomes     40 / 48

 6 / 8 becomes     36 / 48

 Since 40 is bigger than 36, then   40 / 48 >     36 / 48

 Therefore,   5 / 6 >    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.

That is
5 / 10

If you eat one more, that is 6 slices

That is
6 / 10

Now it has become obvious that

 6 / 10 >   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
2 / 10

If you grab 2 slices from the second, the expression for the fraction
2 / 15

Slices for the latter will definitely smaller.

 Therefore,   2 / 10 >    2 / 15

This lesson about comparing fractions is over.

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