Sam surveyed his classmates for a class assignment. He asked his classmates about some of their interests and hobbies. The results of the survey were:

watched TV

rode bikes

read books

played sports

\begin{align*}\frac{3}{8}\end{align*} played an instrument or sang

Sam is using this information to make a presentation. What is the order of activities if Sam were to rank them from the most popular to least?

In this concept, you will learn to order fractions using lowest common denominators.

### Using the Least Common Denominator to Order Fractions

Sometimes, you will need to write fractions in **order** from least to greatest or from greatest to least. This becomes very simple if the fractions have the same denominator.

Write in order from least to greatest.

Since all of these fractions have a common denominator, use the numerators and arrange them in order from the smallest numerator to the largest numerator.

The answer is

To order fractions that do not have a common denominator, rewrite all of the fractions using the **lowest common denominator** (LCD).

Order these fractions from least to greatest.

First, find the LCD. The lowest common multiple of 3, 4, 2, and 6 is 12. Remember, if you cannot figure out the LCD in your head, list the multiples to find the **least common multiple** (LCM).

Then, rewrite each fraction with the denominator 12.

\begin{align*}\begin{array}{rcl} \frac{2}{3} & = & \frac{8}{12}\\ \frac{1}{4} & = & \frac{3}{12}\\ \frac{1}{2} & = & \frac{6}{12}\\ \frac{5}{6} & = & \frac{10}{12} \end{array}\end{align*}

Next, order the fractions from least to greatest.

\begin{align*}\frac{1}{4}, \frac{1}{2}, \frac{2}{3}, \frac{5}{6}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Sam’s survey.

Sam wants to find the order of activities from most popular to least popular. Compare the fractions and list them from greatest to least.

watched TV

rode bikes

read books

played sports

played an instrument or sang

First, find the LCD. The LCD of 4 and 8 is 8.

Then, rewrite each fraction with the denominator of 8. Three of the fractions already have denominators of 8. Find the equivalent fraction for

and .

Next, order the fractions from greatest to least.

Sam should list the activities as:

Most Popular to Least Popular Activity

- Watch TV
- Play sports
- Read books
- Ride bike
- Play instrument or sing

#### Example 2

Write the following fractions in order from least to greatest.

First, find the lowest common denominator. The lowest common multiple of 3 and 7 is 21.

Then, rewrite each fraction with the denominator 12.

Next, order the fractions from least to greatest.

Notice that the original order was in order from least to greatest.

#### Example 3

What would be the LCD for fractions with the denominators of 3, 5, and 6?

The LCD would be 30.

#### Example 4

Rewrite the fractions with a common denominator.

The LCD is 15.

#### Example 5

Write the fractions above in order from greatest to least.

### Review

Write each series in order from least to greatest.