Arrange the following rational numbers in descending order :

(i) $\dfrac{11}{12}, \dfrac{13}{18}, \dfrac{5}{6}, \dfrac{7}{9}$

(ii) $\dfrac{-11}{20}, \dfrac{3}{-10}, \dfrac{17}{-30}, \dfrac{-7}{15}$

(iii) $\dfrac{9}{-24}, -1, \dfrac{2}{-3}, \dfrac{-7}{-6}$

(iv) $\dfrac{7}{-10}, \dfrac{11}{15}, \dfrac{-17}{-30}, \dfrac{-2}{5}$

(i) We have:

$\dfrac{11}{12}, \dfrac{13}{18}, \dfrac{5}{6}, \dfrac{7}{9}$

The L.C.M. of denominators 12, 18, 6, and 9 is 36.

Now, expressing each fraction with denominator 36:

$\dfrac{11}{12} = \dfrac{11 \times 3}{12 \times 3} = \dfrac{33}{36} \\[1em] \dfrac{13}{18} = \dfrac{13 \times 2}{18 \times 2} = \dfrac{26}{36} \\[1em] \dfrac{5}{6} = \dfrac{5 \times 6}{6 \times 6} = \dfrac{30}{36} \\[1em] \dfrac{7}{9} = \dfrac{7 \times 4}{9 \times 4} = \dfrac{28}{36}$

Clearly, $\dfrac{33}{36} \gt \dfrac{30}{36} \gt \dfrac{28}{36} \gt \dfrac{26}{36}$. Therefore $\dfrac{11}{12} \gt \dfrac{5}{6} \gt \dfrac{7}{9} \gt \dfrac{13}{18}$.

Hence, the descending order is: $\dfrac{11}{12}, \dfrac{5}{6}, \dfrac{7}{9}, \dfrac{13}{18}$.

(ii) We have:

$\dfrac{-11}{20}, \dfrac{3}{-10}, \dfrac{17}{-30}, \dfrac{-7}{15}$

First, express each with a positive denominator: $\dfrac{-11}{20}, \dfrac{-3}{10}, \dfrac{-17}{30}, \dfrac{-7}{15}$.

The L.C.M. of denominators 20, 10, 30, and 15 is 60.

Now, expressing each fraction with denominator 60:

$\dfrac{-11}{20} = \dfrac{-11 \times 3}{20 \times 3} = \dfrac{-33}{60} \\[1em] \dfrac{-3}{10} = \dfrac{-3 \times 6}{10 \times 6} = \dfrac{-18}{60} \\[1em] \dfrac{-17}{30} = \dfrac{-17 \times 2}{30 \times 2} = \dfrac{-34}{60} \\[1em] \dfrac{-7}{15} = \dfrac{-7 \times 4}{15 \times 4} = \dfrac{-28}{60}$

Clearly, $\dfrac{-18}{60} \gt \dfrac{-28}{60} \gt \dfrac{-33}{60} \gt \dfrac{-34}{60}$. Therefore $\dfrac{-3}{10} \gt \dfrac{-7}{15} \gt \dfrac{-11}{20} \gt \dfrac{-17}{30}$.

Hence, the descending order is: $\dfrac{3}{-10}, \dfrac{-7}{15}, \dfrac{-11}{20}, \dfrac{17}{-30}$.

(iii) We have:

$\dfrac{9}{-24}, -1, \dfrac{2}{-3}, \dfrac{-7}{-6}$

Expressing with positive denominators: $\dfrac{-9}{24}, \dfrac{-1}{1}, \dfrac{-2}{3}, \dfrac{7}{6}$.

The L.C.M. of denominators 24, 1, 3, and 6 is 24.

Now, expressing each fraction with denominator 24:

$\dfrac{-9}{24} = \dfrac{-9 \times 1}{24 \times 1} = \dfrac{-9}{24} \\[1em] -1 = \dfrac{-1 \times 24}{1 \times 24} = \dfrac{-24}{24} \\[1em] \dfrac{-2}{3} = \dfrac{-2 \times 8}{3 \times 8} = \dfrac{-16}{24} \\[1em] \dfrac{7}{6} = \dfrac{7 \times 4}{6 \times 4} = \dfrac{28}{24}$

Clearly, $\dfrac{28}{24} \gt \dfrac{-9}{24} \gt \dfrac{-16}{24} \gt \dfrac{-24}{24}$. Therefore $\dfrac{7}{6} \gt \dfrac{-9}{24} \gt \dfrac{-2}{3} \gt -1$.

Hence, the descending order is: $\dfrac{-7}{-6}, \dfrac{9}{-24}, \dfrac{2}{-3}, -1$.

(iv) We have:

$\dfrac{7}{-10}, \dfrac{11}{15}, \dfrac{-17}{-30}, \dfrac{-2}{5}$

Expressing with positive denominators: $\dfrac{-7}{10}, \dfrac{11}{15}, \dfrac{17}{30}, \dfrac{-2}{5}$.

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

Now, expressing each fraction with denominator 30:

$\dfrac{-7}{10} = \dfrac{-7 \times 3}{10 \times 3} = \dfrac{-21}{30} \\[1em] \dfrac{11}{15} = \dfrac{11 \times 2}{15 \times 2} = \dfrac{22}{30} \\[1em] \dfrac{17}{30} = \dfrac{17 \times 1}{30 \times 1} = \dfrac{17}{30} \\[1em] \dfrac{-2}{5} = \dfrac{-2 \times 6}{5 \times 6} = \dfrac{-12}{30}$

Clearly, $\dfrac{22}{30} \gt \dfrac{17}{30} \gt \dfrac{-12}{30} \gt \dfrac{-21}{30}$. Therefore $\dfrac{11}{15} \gt \dfrac{17}{30} \gt \dfrac{-2}{5} \gt \dfrac{-7}{10}$.

Hence, the descending order is: $\dfrac{11}{15}, \dfrac{-17}{-30}, \dfrac{-2}{5}, \dfrac{7}{-10}$.