Comparing and Ordering Fractions

To compare and order fractions:

(1) Find the Least Common Multiple (LCM) of the denominators

(2) Rewrite each fraction as an equivalent fraction whose denominator is the LCM

(3) Compare the numerators

A. Comparing Fractions

Compare each pair of fractions using the symbols: >, <, or =

Example 1

\begin{array}{cccc} 3\dfrac{1}{5}& \:\:\dotso\:\: &\:\:\dfrac{37}{10}\:\:\\\\ \color{green}\left(\dfrac{15}{5} + \dfrac{1}{5}\right) & \:\:\dotso\:\: & \color{green}\:\:\dfrac{37}{10}\:\:\\\\ \color{green}\dfrac{16}{5} & \:\:\dotso\:\: & \color{green}\:\:\dfrac{37}{10}\:\:\\\\ \color{green}\dfrac{16 \times \color{red}2}{5\times \color{red}2} &\:\:\dotso\:\: &\color{green} \:\: \dfrac{37}{10}\:\:\\\\ \color{green}\dfrac{32}{10} & \:\:<\:\: & \color{green}\:\:\dfrac{37}{10}\:\:\\\\ \end{array}

Example 2

\begin{array}{cccc} 3.75\%& \:\:\dotso\:\: &\:\:\dfrac{3}{125}\:\:\\\\ \color{green}\dfrac{3.75}{100}& \:\:\dotso\:\: & \color{green}\:\:\dfrac{3}{125}\:\:\\\\ \color{green}\dfrac{3.75 \times \color{red} 4}{100 \times \color{red} 4} & \:\:\dotso\:\: & \color{green}\:\:\dfrac{3 \times \color{red} 4}{10 \times \color{red} 4}\:\:\\\\ \color{green}\dfrac{15}{400} &\:\:>\:\:&\color{green} \:\: \dfrac{12}{400}\:\:\\\\ \end{array}

B. Ordering Fractions

Arrange the following fractions in ascending order

\begin{array}{ccccc} \dfrac{3}{7} & \dfrac{2}{3}&\dfrac{7}{12}&\dfrac{1}{6} \end{array}

Step 1: Find the Least Common Multiple (LCM) of the denominators

\begin{equation*} \begin{split} \text{Prime factorization of 7}& = 7^1\\\\ \text{Prime factorization of 3}& = 3^1\\\\ \text{Prime factorization of 12}& = 2^2 \times 3^1\\\\ \text{Prime factorization of 6}& = 2^1 \times 3^1\\\\ \text{So, the LCM of 7, 3, 12, and 6} & = 2^2 \times 3^1 \times 7^1 = 84 \end{split} \end{equation*}

Step 2: Rewrite each fraction as an equivalent fraction whose denominator is 84

\begin{array}{cccccc} \dfrac{3 \times \color{red}12}{7\times \color{red}12} & \dfrac{2 \times \color{red}28}{3 \times \color{red}28}&\dfrac{7 \times \color{red}7}{12\times \color{red}7}&\dfrac{1\times \color{red}14}{6 \times \color{red}14}&\\\\ \dfrac{36}{84} & \dfrac{56}{84}&\dfrac{49}{84}&\dfrac{14}{84}&\\\\ \color{red}2^{\text{nd}} & \color{red}4^{\text{th}}&\color{red} 3^{\text{rd}}&\color{red}1^{\text{st}}&\\\\ \dfrac{14}{84} & \dfrac{36}{84}&\dfrac{49}{84}&\dfrac{56}{84}&\color{blue}\text{in ascending order}\\\\ \dfrac{1}{6} & \dfrac{3}{7}&\dfrac{7}{12}&\dfrac{2}{3}&\\\\ \end{array}