Have you ever been to an ice cream social? Well, the sixth grade is having one. Terrence and Emilia estimated that one - third of the class will want to eat vanilla ice cream and four - sevenths of the class will want to eat chocolate ice cream. They haven't even gotten to the toppings yet. If these two estimates are accurate, which flavor of ice cream will be the most popular?

**
This is where you come in. You will need to understand how to compare fractions, and that is what you will learn in this Concept.
**

### Guidance

Previously we worked on
**
equivalent fractions
**
, you may have noticed that some fractions have different

**. Remember that when we are talking about the denominator we are talking about the bottom number of the fraction. The**

*denominators***refers to the top number of the fraction.**

*numerator*
\begin{align*}\frac{1}{4}\end{align*}
*
and
*
\begin{align*}\frac{2}{3}\end{align*}

Here we have two different fractions with two different denominators. Remember that the denominator lets us know how many parts one whole has been divided into. Here the first fraction, one-fourth, has been divided into four parts. The second fraction, two-thirds, has been divided into three parts. In this example, we have two different fractions to compare.

**
How do we compare fractions?
**

When we compare two fractions, we want to figure out which fraction is larger and which one is smaller. If we have two fractions with the same denominator, it becomes easier to determine which fraction is greater and which one is less.

\begin{align*}\frac{1}{5} {\underline{\;\;\;\;\;\;\;\;\;}} \frac{3}{5}\end{align*}

We want to use greater than >, less than < or equal to = to compare these two fractions. This one is easy because our denominators are the same.
**
They have common or like denominators.
**
Think about this in terms of pizza.

If both pizzas were divided into five pieces and one person has one-fifth of the pizza and the other person has three-fifths of the pizza, who has more pizza? The person with three-fifths of the pizza has more pizza. Therefore, we can compare those fractions like this.

\begin{align*}\frac{1}{5} < \frac{3}{5}\end{align*}

**
How do we compare fractions that do not have common or like denominators?
**

**
When we are trying to compare two fractions that do not have like denominators, it helps to rewrite them so that they have a common denominator.
**

Let’s look at the two fractions we had earlier.

\begin{align*}\frac{1}{4} \underline{\;\;\;\;\;\;\;\;} \frac{2}{3}\end{align*}

We want to compare these fractions, but that is difficult because we have two different denominators. We can rewrite the denominators by finding the least common multiple of each denominator.
**
This least common multiple becomes the lowest common denominator.
**

**
First, write out the multiples of 4 and 3.
**

**
4, 8, 12
**

**
3, 6, 9, 12
**

**
I can stop there because twelve is the lowest common denominator of both 4 and 3.
**
**
Next, we rewrite the each fraction in terms of twelfths. This means we make an a fraction equivalent to one-fourth in terms of twelfths, and we make a fraction equivalent to two-thirds in terms of twelfths.
**

\begin{align*}\frac{1}{4} = \frac{}{12}\end{align*}

Remember back to creating equal fractions? We multiplied the numerator and the denominator by the same number to create the equal fraction. Well, half of our work is done for us here. Four times three is twelve. We need to complete our equal fraction by multiplying the numerator by 3 too.

\begin{align*}\frac{1}{4} = \frac{3}{12}\end{align*}

Now we can work on rewriting two-thirds in terms of twelfths.

\begin{align*}\frac{2}{3} = \frac{8}{12}\end{align*}

**
Now that both fractions have been written in terms of twelfths, we can compare them.
**

\begin{align*}\frac{3}{12} < \frac{8}{12}\end{align*}

*
so
*

\begin{align*}\frac{1}{4} < \frac{2}{3}\end{align*}

Now it is time for you to practice. Rewrite each with a lowest common denominator and compare using <, >, or =.

#### Example A

\begin{align*}\frac{2}{5} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{6}{10}\end{align*}

**
Solution: <
**

#### Example B

\begin{align*}\frac{2}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{1}{9}\end{align*}

**
Solution: >
**

#### Example C

\begin{align*}\frac{3}{4} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{6}{8}\end{align*}

**
Solution: =
**

Now let's think about the ice cream social. Have you figured out how to compare those fractions?

We need to figure out which flavor will be more popular based on the estimates.

Terrence and Emilia estimated that one - third of the class will want to eat vanilla ice cream and four - sevenths of the class will want to eat chocolate ice cream.

We need to compare one - third with four - sevenths.

\begin{align*}\frac{1}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{4}{7}\end{align*}

To accurately compare these fractions, we need to find the lowest common denominator of each fraction.

The lowest common denominator of 3 and 7 is 21.

Now we can rename these fractions.

\begin{align*}\frac{7}{21} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{12}{21}\end{align*}

**
The chocolate ice cream is more popular than the vanilla.
**

### Vocabulary

- Equivalent Fractions
- two equal fractions

- Denominator
- the bottom number of a fraction

- Numerator
- the top number of a fraction

- Like Denominator
- when two or more denominators are the same, can also be called common denominators.

- Lowest Common Denominator
- the least common multiple of two or more denominators.

### Guided Practice

Here is one for you to try on your own.

Compare the following fractions.

\begin{align*}\frac{6}{9} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{3}{4}\end{align*}

**
Answer
**

To do this, we have to rename both fractions in terms of a lowest common denominator.

The lowest common denominator between 9 and 4 is 36.

\begin{align*}\frac{24}{36} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{27}{36}\end{align*}

**
Our answer is less than.
**

### Video Review

- http://www.mathplayground.com/howto_comparefractions.html – This is a great video on comparing and ordering fractions. The information is presented very clearly.

### Practice

Directions: Rename each in terms of tenths.

1. \begin{align*}\frac{1}{5}\end{align*}

2. \begin{align*}\frac{3}{5}\end{align*}

3. \begin{align*}\frac{1}{2}\end{align*}

4. \begin{align*}\frac{4}{5}\end{align*}

Directions: Complete each equal fraction.

5. \begin{align*}\frac{1}{3} = \frac{}{\;9\;}\end{align*}

6. \begin{align*}\frac{2}{3} = \frac{}{18}\end{align*}

7. \begin{align*}\frac{5}{6} = \frac{}{18}\end{align*}

8. \begin{align*}\frac{2}{7} = \frac{}{14}\end{align*}

9. \begin{align*}\frac{4}{9} = \frac{}{36}\end{align*}

10. \begin{align*}\frac{3}{4} = \frac{}{48}\end{align*}

Directions: Identify the lowest common denominator for each pair of numbers.

11. 3 and 6

12. 4 and 10

13. 5 and 3

14. 7 and 2

15. 8 and 4

16. 6 and 4

17. 8 and 5

18. 12 and 5

19. 9 and 2

20. 6 and 7

Directions: Compare the following fractions using <, >, or =

21. \begin{align*}\frac{1}{2} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{1}{3}\end{align*}

22. \begin{align*}\frac{2}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{3}{9}\end{align*}

23. \begin{align*}\frac{4}{6} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{2}{3}\end{align*}

24. \begin{align*}\frac{6}{10} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{4}{5}\end{align*}

25. \begin{align*}\frac{9}{18} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{3}{6}\end{align*}