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:
| 25 | and 36 |
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.
| 26 | , 16 | and 56 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 23 | and 34 |
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 23 | and 9 beneath 34 |
| 23 | 34 |
| 8 | 9 |
| Since 8 is smaller than 9, then 23 | < 34 |
| Let us compare 56 | and 68 |
5 × 8 = 40 and 6 × 6 = 36
| Put 40 beneath 56 | and 36 beneath 68 |
| 56 | 68 |
| 40 | 36 |
| Since 40 is bigger than 36, then 56 | > 68 |
A second method to use when comparing fractions is to first get a common denominator.
| Let us compare again 56 | and 68 |
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.
| 56 | becomes 4048 |
| 68 | becomes 3648 |
| Since 40 is bigger than 36, then 4048 | > 3648 |
| Therefore, 56 | > 68 |
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
510
If you eat one more, that is 6 slices
That is
610
Now it has become obvious that
| 610 | > 510 |
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
210
If you grab 2 slices from the second, the expression for the fraction
215
Slices for the latter will definitely smaller.
| Therefore, 210 | > 215 |
This lesson about comparing fractions is over.
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