Arrange the following rational numbers in ascending order :

(i) $\dfrac{3}{4}, \dfrac{5}{8}, \dfrac{11}{16}, \dfrac{21}{32}$

(ii) $\dfrac{-2}{5}, \dfrac{7}{-10}, \dfrac{-8}{15}, \dfrac{17}{-30}$

(iii) $\dfrac{5}{-12}, \dfrac{-2}{3}, \dfrac{-7}{9}, \dfrac{11}{-18}$

(iv) $\dfrac{-4}{7}, \dfrac{13}{-28}, \dfrac{9}{14}, \dfrac{23}{42}$

(i) We have:

$\dfrac{3}{4}, \dfrac{5}{8}, \dfrac{11}{16}, \dfrac{21}{32}$

First we find the L.C.M.

L.C.M. of denominators 4, 8, 16, and 32 is 32.

Now, expressing each fraction with denominator 32:

$\dfrac{3}{4} = \dfrac{3 \times 8}{4 \times 8} = \dfrac{24}{32} \\[1em] \dfrac{5}{8} = \dfrac{5 \times 4}{8 \times 4} = \dfrac{20}{32} \\[1em] \dfrac{11}{16} = \dfrac{11 \times 2}{16 \times 2} = \dfrac{22}{32} \\[1em] \dfrac{21}{32} = \dfrac{21 \times 1}{32 \times 1} = \dfrac{21}{32} \\[1em]$

Clearly, $\dfrac{20}{32} \lt \dfrac{21}{32} \lt \dfrac{22}{32} \lt \dfrac{24}{32}$. Therefore $\dfrac{5}{8} \lt \dfrac{21}{32} \lt \dfrac{11}{16} \lt \dfrac{3}{4}$

Hence, the ascending order is: $\dfrac{5}{8}, \dfrac{21}{32}, \dfrac{11}{16}, \dfrac{3}{4}$.

(ii) We have:

$\dfrac{-2}{5}, \dfrac{7}{-10}, \dfrac{-8}{15}, \dfrac{17}{-30}$

First, express each rational number with a positive denominator: $\dfrac{-2}{5}, \dfrac{-7}{10}, \dfrac{-8}{15}, \dfrac{-17}{30}$.

The L.C.M. of denominators 5, 10, 15, and 30 is 30.

Now, expressing each fraction with denominator 30:

$\dfrac{-2}{5} = \dfrac{-2 \times 6}{5 \times 6} = \dfrac{-12}{30} \\[1em] \dfrac{-7}{10} = \dfrac{-7 \times 3}{10 \times 3} = \dfrac{-21}{30} \\[1em] \dfrac{-8}{15} = \dfrac{-8 \times 2}{15 \times 2} = \dfrac{-16}{30} \\[1em] \dfrac{-17}{30} = \dfrac{-17 \times 1}{30 \times 1} = \dfrac{-17}{30}$

Clearly, $\dfrac{-21}{30} \lt \dfrac{-17}{30} \lt \dfrac{-16}{30} \lt \dfrac{-12}{30}$. Therefore $\dfrac{-7}{10} \lt \dfrac{-17}{30} \lt \dfrac{-8}{15} \lt \dfrac{-2}{5}$

Hence, the ascending order is: $\dfrac{7}{-10}, \dfrac{17}{-30}, \dfrac{-8}{15}, \dfrac{-2}{5}$.

(iii) We have:

$\dfrac{5}{-12}, \dfrac{-2}{3}, \dfrac{-7}{9}, \dfrac{11}{-18}$

Expressing with positive denominators: $\dfrac{-5}{12}, \dfrac{-2}{3}, \dfrac{-7}{9}, \dfrac{-11}{18}$.

The L.C.M. denominators of 12, 3, 9, and 18 is 36.

Now, expressing each fraction with denominator 36:

$\dfrac{-5}{12} = \dfrac{-5 \times 3}{12 \times 3} = \dfrac{-15}{36} \\[1em] \dfrac{-2}{3} = \dfrac{-2 \times 12}{3 \times 12} = \dfrac{-24}{36} \\[1em] \dfrac{-7}{9} = \dfrac{-7 \times 4}{9 \times 4} = \dfrac{-28}{36} \\[1em] \dfrac{-11}{18} = \dfrac{-11 \times 2}{18 \times 2} = \dfrac{-22}{36}$

Clearly, $\dfrac{-28}{36} \lt \dfrac{-24}{36} \lt \dfrac{-22}{36} \lt \dfrac{-15}{36}$. Therefore $\dfrac{-7}{9} \lt \dfrac{-2}{3} \lt \dfrac{-11}{18} \lt \dfrac{-5}{12}$.

Hence, the ascending order is: $\dfrac{-7}{9}, \dfrac{-2}{3}, \dfrac{11}{-18}, \dfrac{5}{-12}$.

(iv) We have:

$\dfrac{-4}{7}, \dfrac{13}{-28}, \dfrac{9}{14}, \dfrac{23}{42}$

Expressing with positive denominators: $\dfrac{-4}{7}, \dfrac{-13}{28}, \dfrac{9}{14}, \dfrac{23}{42}$.

The L.C.M. of denominators 7, 28, 14, and 42 is 84.

Now, expressing each fraction with denominator 84:

$\dfrac{-4}{7} = \dfrac{-4 \times 12}{7 \times 12} = \dfrac{-48}{84} \\[1em] \dfrac{-13}{28} = \dfrac{-13 \times 3}{28 \times 3} = \dfrac{-39}{84} \\[1em] \dfrac{9}{14} = \dfrac{9 \times 6}{14 \times 6} = \dfrac{54}{84} \\[1em] \dfrac{23}{42} = \dfrac{23 \times 2}{42 \times 2} = \dfrac{46}{84}$

Clearly, $\dfrac{-48}{84} \lt \dfrac{-39}{84} \lt \dfrac{46}{84} \lt \dfrac{54}{84}$. Therefore $\dfrac{-4}{7} \lt \dfrac{-13}{28} \lt \dfrac{23}{42} \lt \dfrac{9}{14}$.

Hence, the ascending order is: $\dfrac{-4}{7}, \dfrac{13}{-28}, \dfrac{23}{42}, \dfrac{9}{14}$.