5.4: Comparing Fractions

Learning Objectives

• understand ordering of numbers and be familiar with grouping symbols
• be able to compare two or more fractions

Order and the Inequality Symbols

Our number system is called an ordered number system because the numbers in the system can be placed in order from smaller to larger. This is easily seen on the number line.

On the number line, a number that appears to the right of another number is larger than that other number. For example, 5 is greater than 2 because 5 is located to the right of 2 on the number line. We may also say that 2 is less than 5.

To make the inequality phrases "greater than" and "less than" more brief, mathematicians represent them with the symbols > and <, respectively.

Symbols for Greater Than > and Less Than <
> represents the phrase "greater than."
< represents the phrase "less than."

5 > 2 represents "5 is greater than 2."

2 < 5 represents "2 is less than 5."

Comparing Fractions

Recall that the fraction \(\dfrac{4}{5}\) indicates that we have 4 of 5 parts of some whole quantity, and the fraction \(\dfrac{3}{5}\) indicates that we have 3 of 5 parts. Since 4 of 5 parts is more than 3 of 5 parts, \(\dfrac{4}{5}\) is greater than \(\dfrac{3}{5}\); that is,

\(\dfrac{4}{5} > \dfrac{3}{5}\)

We have just observed that when two fractions have the same denominator, we can determine which is larger by comparing the numerators.

Comparing Fractions
If two fractions have the same denominators, the fraction with the larger nu­merator is the larger fraction.

Thus, to compare the sizes of two or more fractions, we need only convert each of them to equivalent fractions that have a common denominator. We then compare the numerators. It is convenient if the common denominator is the LCD. The fraction with the larger numerator is the larger fraction.

Sample Set A

Compare \(\dfrac{8}{9}\) and \(\dfrac{14}{15}\). Convert each fraction to an equivalent fraction with the LCD as the denominator. Find the LCD.

Solution

\(\left \{ \begin{array} {rcl} {9} & = & {3^2} \\ {15} & = & {3 \cdot 5} \end{array} \right \} \text{ The LCD } = 3^2 \cdot 5 = 9 \cdot 5 = 45\)

\(\dfrac{8}{9} = \dfrac{8 \cdot 5}{45} = \dfrac{40}{45}\)

\(\dfrac{14}{15} = \dfrac{14 \cdot 3}{45} = \dfrac{42}{45}\)

Since \(40 < 42\).

\(\dfrac{40}{45} < \dfrac{42}{45}\)

Thus \(\dfrac{8}{9} < \dfrac{14}{15}\).

Sample Set A

Write \(\dfrac{5}{6}\), \(\dfrac{7}{10}\), and \(\dfrac{13}{15}\) in order from smallest to largest.

Convert each fraction to an equivalent fraction with the LCD as the denominator.

Find the LCD.

Solution

\(\left \{ \begin{array} {rcl} {6} & = & {2 \cdot 3} \\ {10} & = & {2 \cdot 5} \\ {15} & = & {3 \cdot 5} \end{array} \right \} \text{ The LCD } = 2 \cdot 3 \cdot 5 = 30\)

\(\dfrac{5}{6} = \dfrac{5 \cdot 5}{30} = \dfrac{25}{30}\)

\(\dfrac{7}{10} = \dfrac{7 \cdot 3}{30} = \dfrac{21}{30}\)

\(\dfrac{13}{15} = \dfrac{13 \cdot 2}{30} = \dfrac{26}{30}\)

Since \(21 < 25 < 26\).

\(\dfrac{21}{30} < \dfrac{25}{30} < \dfrac{26}{30}\)

\(\dfrac{7}{10} < \dfrac{5}{6} < \dfrac{13}{15}\)

Writing these numbers in order from smallest to largest, we get \(\dfrac{7}{10}\), \(\dfrac{5}{6}\), \(\dfrac{13}{15}\).

Sample Set A

Compare \(8 \dfrac{6}{7}\) and \(6 \dfrac{3}{4}\).

Solution

To compare mixed numbers that have different whole number parts, we need only compare whole number parts. Since 6 < 8, \(6 \dfrac{3}{4} < 8 \dfrac{6}{7}\)

Sample Set A

Compare \(4 \dfrac{5}{8}\) and \(4 \dfrac{7}{12}\)

Solution

To compare mixed numbers that have the same whole number parts, we need only compare fractional parts.

\(\left \{ \begin{array} {rcl} {8} & = & {2 \cdot 3} \\ {12} & = & {2^2 \cdot 3} \end{array} \right \} \text{ The LCD } = 2^3 \cdot 3 = 8 \cdot 3 = 24\)

\(\dfrac{5}{8} = \dfrac{5 \cdot 3}{24} = \dfrac{15}{24}\)

\(\dfrac{7}{12} = \dfrac{7 \cdot 2}{24} = \dfrac{14}{24}\)

Since \(14 < 15\).

\(\dfrac{14}{24} < \dfrac{15}{24}\)

\(\dfrac{7}{12} < \dfrac{5}{8}\)

Hence, \(4 \dfrac{7}{12} < 4 \dfrac{5}{8}\)

Practice Set A

Compare \(\dfrac{3}{4}\) and \(\dfrac{4}{5}\)

Answer

\(\dfrac{3}{4} < \dfrac{4}{5}\)

Practice Set A

Compare \(\dfrac{9}{10}\) and \(\dfrac{13}{15}\)

Answer

\(\dfrac{13}{15} < \dfrac{9}{10}\)

Practice Set A

Write \(\dfrac{13}{16}\), \(\dfrac{17}{20}\), and \(\dfrac{33}{40}\) in order from smallest to largest.

Answer

\(\dfrac{13}{16}\), \(\dfrac{33}{40}\), \(\dfrac{17}{20}\)

Practice Set A

Compare \(11 \dfrac{1}{6}\) and \(9 \dfrac{2}{5}\).

Answer

\(9 \dfrac{2}{5} < 11 \dfrac{1}{6}\)

Practice Set A

Compare \(1 \dfrac{9}{14}\) and \(1 \dfrac{11}{16}\).

Answer

\(1 \dfrac{9}{14} < 1 \dfrac{11}{16}\)

Exercises

Arrange each collection of numbers in order from smallest to largest.

Exercise \(\PageIndex{1}\)

\(\dfrac{3}{5}\), \(\dfrac{5}{8}\)

Answer

\(\dfrac{3}{5} < \dfrac{5}{8}\)

Exercise \(\PageIndex{2}\)

\(\dfrac{1}{6}, \dfrac{2}{7}\)

Exercise \(\PageIndex{3}\)

\(\dfrac{3}{4}\), \(\dfrac{5}{6}\)

Answer

\(\dfrac{3}{4} < \dfrac{5}{6}\)

Exercise \(\PageIndex{4}\)

\(\dfrac{7}{9}, \dfrac{11}{12}\)

Exercise \(\PageIndex{5}\)

\(\dfrac{3}{8}\), \(\dfrac{2}{5}\)

Answer

\(\dfrac{3}{8} < \dfrac{2}{5}\)

Exercise \(\PageIndex{6}\)

\(\dfrac{1}{2}, \dfrac{5}{8}, \(\dfrac{7}{16}\)

Exercise \(\PageIndex{7}\)

\(\dfrac{1}{2}\), \(\dfrac{3}{5}\), \(\dfrac{4}{7}\)

Answer

\(\dfrac{1}{2} < \dfrac{4}{7} < \dfrac{3}{5}\)

Exercise \(\PageIndex{8}\)

\(\dfrac{3}{4}, \dfrac{2}{3}, \(\dfrac{5}{6}\)

Exercise \(\PageIndex{9}\)

\(\dfrac{3}{4}\), \(\dfrac{7}{9}\), \(\dfrac{5}{4}\)

Answer

\(\dfrac{3}{4} < \dfrac{7}{9} < \dfrac{5}{4}\)

Exercise \(\PageIndex{10}\)

\(\dfrac{7}{8}, \dfrac{15}{16}, \(\dfrac{11}{12}\)

Exercise \(\PageIndex{11}\)

\(\dfrac{3}{14}\), \(\dfrac{2}{7}\), \(\dfrac{3}{4}\)

Answer

\(\dfrac{3}{14} < \dfrac{2}{7} < \dfrac{3}{4}\)

Exercise \(\PageIndex{12}\)

\(\dfrac{17}{32}, \dfrac{25}{48}, \(\dfrac{13}{16}\)

Exercise \(\PageIndex{13}\)

\(5 \dfrac{3}{5}\), \(5 \dfrac{4}{7}\)

Answer

\(5 \dfrac{4}{7} < 5 \dfrac{3}{5}\)

Exercise \(\PageIndex{14}\)

\(11 \dfrac{3}{16}, 11 \dfrac{1}{12}\)

Exercise \(\PageIndex{15}\)

\(9 \dfrac{2}{3}\), \(9 \dfrac{4}{5}\)

Answer

\(9 \dfrac{2}{3} < 9 \dfrac{4}{5}\)

Exercise \(\PageIndex{16}\)

\(7 \dfrac{2}{3}, 8 \dfrac{5}{6}\)

Exercise \(\PageIndex{17}\)

\(1 \dfrac{9}{16}\), \(2 \dfrac{1}{20}\)

Answer

\(1 \dfrac{9}{16} < 2 \dfrac{1}{20}\)

Exercise \(\PageIndex{18}\)

\(20 \dfrac{15}{16}, 20 \dfrac{23}{24}\)

Exercise \(\PageIndex{19}\)

\(2 \dfrac{2}{9}\), \(2 \dfrac{3}{7}\)

Answer

\(2 \dfrac{2}{9} < 2 \dfrac{3}{7}\)

Exercise \(\PageIndex{20}\)

\(5 \dfrac{8}{13}, 5 \dfrac{9}{20}\)

Exercises for Review

Exercise \(\PageIndex{21}\)

Round 267,006,428 to the nearest ten million.

Answer

270,000,000

Exercise \(\PageIndex{22}\)

Is the number 82,644 divisible by 2? by 3? by 4?

Exercise \(\PageIndex{23}\)

Convert \(3 \dfrac{2}{7}\) to an improper fraction.

Answer

\(\dfrac{23}{7}\)

Exercise \(\PageIndex{24}\)

Find the value of \(\dfrac{5}{6} + \dfrac{3}{10} - \dfrac{2}{5}\)

Exercise \(\PageIndex{25}\)

Find the value of \(8 \dfrac{3}{8} + 5 \dfrac{1}{4}\).

Answer

\(13 \dfrac{5}{8}\) or \(\dfrac{109}{8}\)