5.4 Comparing fractions

Compare $\frac{8}{9}$ and $\frac{\text{14}}{\text{15}}$ .

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

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

$\begin{array}{c}\frac{8}{9}=\frac{8\cdot 5}{\text{45}}=\frac{\text{40}}{\text{45}}\hfill \\ \frac{\text{14}}{\text{15}}=\frac{\text{14}\cdot 3}{\text{45}}=\frac{\text{42}}{\text{45}}\hfill \end{array}$

Since ,

Thus .

Write $\frac{5}{6},\frac{7}{\text{10}},$ and $\frac{\text{13}}{\text{15}}$ in order from smallest to largest.

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

Find the LCD.

$\left(\begin{array}{c}\hfill 6=2\cdot 3\\ \hfill \text{10}=2\cdot 5\\ \hfill \text{15}=3\cdot 5\end{array}\right\}\text{The LCD}=2\cdot 3\cdot 5=\text{30}$

$\frac{5}{6}=\frac{5\cdot 5}{\text{30}}=\frac{\text{25}}{\text{30}}$

$\frac{7}{\text{10}}=\frac{7\cdot 3}{\text{30}}=\frac{\text{21}}{\text{30}}$

$\frac{\text{13}}{\text{15}}=\frac{\text{13}\cdot 2}{\text{30}}=\frac{\text{26}}{\text{30}}$

Since $21<25<26$ ,

$\frac{\text{21}}{\text{30}}$ < $\frac{\text{25}}{\text{30}}$ < $\frac{\text{26}}{\text{30}}$

$\frac{7}{\text{10}}$ < $\frac{5}{6}$ < $\frac{\text{13}}{\text{15}}$

Writing these numbers in order from smallest to largest, we get $\frac{7}{\text{10}}$ , $\frac{5}{6}$ , $\frac{\text{13}}{\text{15}}$ .

Compare $8\frac{6}{7}$ and $6\frac{3}{4}$ .

To compare mixed numbers that have different whole number parts, we need only compare whole number parts. Since 6<8,

Compare $4\frac{5}{8}\text{and}4\frac{7}{\text{12}}$

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

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

$\frac{5}{8}=\frac{5\cdot 3}{\text{24}}=\frac{\text{15}}{\text{24}}$

$\frac{7}{\text{12}}=\frac{7\cdot 2}{\text{24}}=\frac{\text{14}}{\text{24}}$

Since 14<15,

Hence,