Question

Arrange the following rational numbers in ascending order:
(i) $\frac{3}{5},\frac{-17}{-30},\frac{8}{-15},\frac{-7}{10}$
(ii) $\frac{-4}{9},\frac{5}{-12},\frac{7}{-18},\frac{2}{-3}$

Answer 1

(i) Ascending order:

$$\begin{gathered} \mathrm{Since},\mathrm{LCM}\hspace{0.17em}\mathrm{of}5,-30,-15,10\mathrm{is}30. \\ \mathrm{Multiplying}\mathrm{the}\mathrm{numerators}\mathrm{and}\mathrm{denominators}\mathrm{to}\mathrm{get}\mathrm{the}\mathrm{denominator}\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{LCM}, \\ \frac{3}{5}=\frac{3\times 6}{5\times 6}=\frac{18}{30}, \\ \frac{17}{30}=\frac{17\times 1}{30\times 1}=\frac{17}{30}, \\ \frac{8}{-15}=\frac{-8\times 2}{15\times 2}=\frac{-16}{30}, \\ \frac{-7}{10}=\frac{-7\times 3}{10\times 3}=\frac{-21}{30}. \\ \mathrm{Order}\mathrm{is}-21<-16<17<8. \\ \mathrm{Order}\mathrm{is}\frac{-7}{10}<\frac{8}{-15}<\frac{17}{30}<\frac{3}{5}. \end{gathered}$$

(ii)

$$\begin{gathered} \mathrm{Since},\mathrm{LCM}\hspace{0.17em}\mathrm{of}9,-12,-18,3\mathrm{is}36. \\ \mathrm{Multiplying}\mathrm{the}\mathrm{numerators}\mathrm{and}\mathrm{denominators}\mathrm{to}\mathrm{get}\mathrm{the}\mathrm{denominator}\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{LCM}, \\ \frac{-4}{9}=\frac{-4\times 4}{9\times 4}=\frac{-16}{36} \\ \frac{5}{-12}=\frac{-5\times 3}{12\times 3}=\frac{-15}{36} \\ \frac{7}{-18}=\frac{-7\times 2}{18\times 2}=\frac{-14}{36} \\ \frac{2}{-3}=\frac{-2\times 12}{3\times 12}=\frac{-24}{36}. \\ \mathrm{Order}\mathrm{is}-24<-16<-15<-14. \\ \mathrm{Order}\mathrm{is}\frac{2}{-3}<\frac{-4}{9}<\frac{5}{-12}<\frac{7}{-18}. \end{gathered}$$