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Fraction Ordering with Lowest Common Denominators

Arranging numerators in ascending order

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Fraction Ordering with Lowest Common Denominators
 
License: CC BY-NC 3.0

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:

\begin{align*}\frac{7}{8}\end{align*} watched TV

\begin{align*}\frac{1}{4}\end{align*} rode bikes

\begin{align*}\frac{5}{8}\end{align*} read books

\begin{align*}\frac{3}{4}\end{align*} 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.

\begin{align*}\frac{4}{9}, \frac{2}{9}, \frac{8}{9}, \frac{3}{9}, \frac{6}{9}\end{align*}

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 \begin{align*}\frac{2}{9}, \frac{3}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}\end{align*}

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.

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

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).

\begin{align*}3 - 3, 6, 9, 12 \qquad 4 - 4, 8, 12 \qquad 2 - 2, 4, 6, 8, 10, 12 \qquad 6 - 6, 12\end{align*}

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.

\begin{align*}\frac{7}{8}\end{align*} watched TV

\begin{align*}\frac{1}{4}\end{align*} rode bikes

\begin{align*}\frac{5}{8}\end{align*} read books

\begin{align*}\frac{3}{4}\end{align*} played sports

\begin{align*}\frac{3}{8}\end{align*} 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 \begin{align*}\frac{1}{4}\end{align*} and \begin{align*}\frac{3}{4}\end{align*}.

\begin{align*}\frac{1}{4} = \frac{4}{8}\end{align*}

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

Next, order the fractions from greatest to least.

\begin{align*}\frac{7}{8}, \frac{6}{8}, \frac{5}{8}, \frac{4}{8}, \frac{3}{8}\end{align*}

Sam should list the activities as:

Most Popular to Least Popular Activity

  1. Watch TV
  2. Play sports
  3. Read books
  4. Ride bike
  5. Play instrument or sing

Example 2

Write the following fractions in order from least to greatest.

\begin{align*}\frac{4}{7}, \frac{2}{3}, \frac{5}{7}\end{align*} 

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

Then, rewrite each fraction with the denominator 12.

\begin{align*}\begin{array}{rcl} \frac{4}{7} & = & \frac{12}{21}\\ \frac{2}{3} & = & \frac{14}{21}\\ \frac{5}{7} & = & \frac{15}{21} \end{array}\end{align*}

Next, order the fractions from least to greatest.

\begin{align*}\frac{4}{7}, \frac{2}{3}, \frac{5}{7}\end{align*}

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?

\begin{align*}3 - 3, 6, 9, 12, 15, 18, 21, 24, 27,30 \qquad 5 - 5, 10, 15, 20, 25, 30 \qquad 6 - 6, 12, 18, 24, 30\end{align*}

The LCD would be 30.

Example 4

Rewrite the fractions with a common denominator.

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

\begin{align*}\frac{24}{60}, \frac{6}{30}, \frac{20}{30}\end{align*}

The LCD is 15.

\begin{align*}\begin{array}{rcl} \frac{4}{5} & = & \frac{12}{15}\\ \frac{1}{5} & = & \frac{3}{15}\\ \frac{2}{3} & = & \frac{10}{15} \end{array}\end{align*}

Example 5

Write the fractions above in order from greatest to least.

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

Review

Write each series in order from least to greatest.

  1. \begin{align*}\frac{5}{6}, \frac{1}{3}, \frac{4}{9}\end{align*}
  2. \begin{align*}\frac{6}{7}, \frac{1}{4}, \frac{2}{3}\end{align*}
  3. \begin{align*}\frac{6}{6}, \frac{4}{5}, \frac{2}{3}\end{align*}
  4. \begin{align*}\frac{1}{2}, \frac{3}{5}, \frac{2}{3}\end{align*}
  5. \begin{align*}\frac{2}{7}, \frac{1}{4}, \frac{3}{6}\end{align*}
  6. \begin{align*}\frac{1}{6}, \frac{2}{9}, \frac{2}{5}\end{align*}
  7. \begin{align*}\frac{4}{16}, \frac{4}{5}, \frac{3}{7}\end{align*}
  8. \begin{align*}\frac{9}{10}, \frac{4}{5}, \frac{3}{4}\end{align*}
  9. \begin{align*}\frac{4}{5}, \frac{1}{2}, \frac{2}{3}\end{align*}
  10. \begin{align*}\frac{9}{11}, \frac{2}{3}, \frac{3}{4}\end{align*}
  11. \begin{align*}\frac{4}{7}, \frac{1}{5}, \frac{3}{8}\end{align*}
  12. \begin{align*}\frac{6}{7}, \frac{1}{3}, \frac{2}{5}\end{align*}
  13. \begin{align*}\frac{7}{8}, \frac{4}{5}, \frac{1}{3}\end{align*}
  14. \begin{align*}\frac{1}{6}, \frac{4}{5}, \frac{2}{4}\end{align*}
  15. \begin{align*}\frac{1}{9}, \frac{4}{7}, \frac{2}{9}, \frac{7}{8}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.12.

Resources

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Vocabulary

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. \frac{5}{8} has denominator 8.

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.

Like Denominators

Two or more fractions have like denominators when their denominators are the same. "Common denominators" is a synonym for "like denominators".

Lowest Common Denominator

The lowest common denominator of multiple fractions is the least common multiple of all of the related denominators.

Numerator

The numerator is the number above the fraction bar in a fraction.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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