Question

Arrange the following ratios in ascending order:

(i) (5 : 6), (8 : 9), (11 : 18)
(ii) (11 : 14), (17 : 21), (5 : 7) and (2 : 3)

Answer 1

(i) We have $\frac{5}{6},\frac{8}{9}\mathrm{and}\frac{11}{18}.$
$$\begin{gathered} 2\underline{\overline{)6,9,18}} \\ \underline{3\underline{\overline{)3,9}},9} \\ 3\overline{)\underline{1,3,3}} \\ \overline{)\underline{1,1,1}} \end{gathered}$$

The LCM of 6, 9 and 18 is 18. Therefore, we have:

$$\begin{gathered} \frac{5}{6}=\frac{5\times 3}{6\times 3}=\frac{15}{18} \\ \frac{8}{9}=\frac{8\times 2}{9\times 2}=\frac{16}{18}\frac{11}{18}=\frac{11}{18}\mathrm{Clearly},\frac{11}{18}<\frac{15}{18}<\frac{16}{18} \end{gathered}$$

Hence, (11 : 18) $<$ (5 : 6) $<$ (8 : 9)

(ii) $\mathrm{We}\mathrm{have}\frac{11}{14},\frac{17}{21},\frac{5}{7}\mathrm{and}\frac{2}{3}.$
$$\begin{gathered} 2\underline{\overline{)14,21,7,3}} \\ \underline{7\underline{\overline{)7,21,7,3}},} \\ 3\overline{)\underline{1,3,1,3}} \\ \overline{)\underline{1,1,1,1}} \end{gathered}$$

The LCM of 14, 21, 7 and 3 is 42.

$$\begin{gathered} \frac{11}{14}=\frac{11\times 3}{14\times 3}=\frac{33}{28} \\ \frac{17}{21}=\frac{17\times 2}{21\times 2}=\frac{34}{42} \\ \frac{5}{7}=\frac{5\times 6}{7\times 6}=\frac{30}{42} \\ \frac{2}{3}=\frac{2\times 14}{3\times 14}=\frac{28}{42} \\ \mathrm{Clearly},\frac{28}{42}<\frac{30}{42}<\frac{33}{28}<\frac{34}{42} \\ \mathrm{Hence},(2:3)<(5:7)<(11:14)<(17:21) \end{gathered}$$