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# Fraction Comparison with Lowest Common Denominators

## LCD of equivalent fractions is LCM of the denominators

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Fraction Comparison with Lowest Common Denominators

The 6th grade class is having an ice cream social. 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. If these two estimates are accurate, which flavor of ice cream will be the most popular?

In this concept, you will learn how to compare fractions using the lowest common denominator.

### Comparing Fractions Using the Lowest Common Denominator

Some fractions have different denominators, the bottom number of a fraction. The numerator refers to the top number of a fraction.

Here are two fractions with different denominators.

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

Remember that the denominator is the number of parts the whole has been divided into. In the first fraction, one-fourth, the whole has been divided into four parts. The second fraction, two-thirds, has been divided into three parts. In this example, you cannot compare the numerators because the parts of each fraction have different values.

You use greater than (>), less than (<), or equal to (=) to compare two fractions. It is easy to compare fractions with the same denominator.

Compare these two fractions.

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

Both fractions represent a whole that is divided into 5 parts. If the fractions were pizzas that were divided into 5 parts, one-fifth of a pizza would be less than with three-fifths of the same pizza. Therefore, you can compare those fractions like this.

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

To compare fractions different denominators, rewrite the fractions so they have a common denominator.

Let’s compare the two fractions from earlier.

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

Rewrite the denominators by finding the least common multiple of each denominator. Remember that the least common multiple (LCM) is the smallest multiple that two numbers have in common. This LCM becomes the lowest common denominator (LCD).

First, list the multiples for 3 and 4 and find the LCM.

4, 8, 12, 16

3, 6, 9, 12

The LCM for 3 and 4 is 12.

Then, rewrite each fraction in terms of twelfths. Make a fraction equivalent to one-fourth in terms of twelfths, and make a fraction equivalent to two-thirds in terms of twelfths.

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

To make equivalent fractions, multiply or divide the numerator and the denominator by the same number to create the equal fraction. 4 is multiplied by 3 to get 12. Complete the equivalent fraction by also multiplying the numerator by 3.

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

Now work on rewriting two-thirds in terms of twelfths. 3 is multiplied by 4 to get 12. Multiply the numerator by 4.

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

Next, compare the fractions now that both fractions have been written in terms of twelfths.

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

so

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

\begin{align*}\frac{1}{4}\end{align*} is less than \begin{align*}\frac{2}{3}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about the ice cream social.

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. Compare the fractions to see which flavor will be more popular.

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

First, find the LCM of 3 and 7. The lowest common denominator will be 21.

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of \begin{align*}\frac{1}{3}\end{align*} by 7. Multiply the numerator and denominator of \begin{align*}\frac{4}{7}\end{align*} by 3.

\begin{align*}\frac{1}{3} = \frac{7}{21}\end{align*}

\begin{align*}\frac{4}{7} = \frac{12}{21}\end{align*}

Next, compare the equivalent fractions.

\begin{align*}\frac{7}{21} < \frac{12}{21}\end{align*}

Chocolate will be more popular.

#### Example 2

Rewrite each fraction with the lowest common denominator and compare using <, >, or =.

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

First, find the LCM of 9 and 4. The lowest common denominator will be 36.

9 = 9, 18, 27, 36

4 = 4, 12, 16, 20, 24, 28, 32, 36

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of \begin{align*}\frac{6}{9}\end{align*} by 4. Multiply the numerator and denominator of \begin{align*}\frac{3}{4}\end{align*} by 9.

\begin{align*}\frac{6}{9} = \frac{24}{36}\end{align*}

\begin{align*}\frac{3}{4} = \frac{27}{36}\end{align*}

Next, compare the equivalent fractions.

\begin{align*}\frac{24}{36} < \frac{27}{36}\end{align*}

\begin{align*}\frac{6}{9}\end{align*} is less than \begin{align*}\frac{3}{4}\end{align*}.

#### Example 3

Compare the fractions.

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

First, find the LCM of 5 and 10. The lowest common denominator will be 10.

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of \begin{align*}\frac{2}{5}\end{align*} by 2. \begin{align*}\frac{6}{10}\end{align*} is already a fraction of tenths.

\begin{align*}\frac{2}{5} = \frac{4}{10}\end{align*}

Next, compare the equivalent fractions.

\begin{align*}\frac{4}{10} < \frac{6}{10}\end{align*}

\begin{align*}\frac{2}{5}\end{align*} is less than \begin{align*}\frac{6}{10}\end{align*}.

#### Example 4

Compare the fractions.

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

First, find the LCM of 3 and 9. The lowest common denominator will be 9.

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of \begin{align*}\frac{2}{3}\end{align*} by 3. \begin{align*}\frac{1}{9}\end{align*} does not change.

\begin{align*}\frac{2}{3} = \frac{6}{9}\end{align*}

Next, compare the equivalent fractions.

\begin{align*}\frac{6}{9} > \frac{1}{9}\end{align*}

\begin{align*}\frac{2}{3}\end{align*} is greater than \begin{align*}\frac{1}{9}\end{align*}.

#### Example 5

Compare the fractions.

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

First, find the LCM of 4 and 8. The lowest common denominator will be 8.

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of \begin{align*}\frac{3}{4}\end{align*} by 2. \begin{align*}\frac{6}{8}\end{align*} does not change.

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

Next, compare the equivalent fractions.

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

\begin{align*}\frac{3}{4}\end{align*} is equal to \begin{align*}\frac{6}{8}\end{align*}.

### Review

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*}

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*}

Identify the lowest common multiple 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

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*}

To see the Review answers, open this PDF file and look for section 5.11.

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### Vocabulary Language: English

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.