Arrange the following ratios in descending order of magnitudes :

(i) (5 : 6), (8 : 9), (13 : 18) and (19 : 24)

(ii) (6 : 7), (13 : 14), (19 : 21) and (23 : 28)

(iii) (7 : 12), (9 : 16), (13 : 20) and (5 : 8)

(i) Given,

(5 : 6), (8 : 9), (13 : 18) and (19 : 24)

We convert them into equivalent like fractions.

L.C.M of 6, 9, 18, 24 is 72.

$\Rightarrow \dfrac{5 \times 12}{6 \times 12} = \dfrac{60}{72} \\[1em] \Rightarrow \dfrac{8 \times 8}{9 \times 8} = \dfrac{64}{72} \\[1em] \Rightarrow \dfrac{13 \times 4}{18 \times 4} = \dfrac{52}{72} \\[1em] \Rightarrow \dfrac{19 \times 3}{24 \times 3} = \dfrac{57}{72} \\[1em] \Rightarrow \dfrac{64}{72} \gt \dfrac{60}{72} \gt \dfrac{57}{72} \gt \dfrac{52}{72}$

8 : 9 > 5 : 6 > 19 : 24 > 13 : 18

Hence, the ratios in descending order are (8 : 9) > (5 : 6) > (19 : 24) > (13 : 18).

(ii) Given,

(6 : 7), (13 : 14), (19 : 21) and (23 : 28)

We convert them into equivalent like fractions.

L.C.M of 7, 14, 21, 28 is 84

$\Rightarrow \dfrac{6 \times 12}{7 \times 12} = \dfrac{72}{84} \\[1em] \Rightarrow \dfrac{13 \times 6}{14 \times 6} = \dfrac{78}{84} \\[1em] \Rightarrow \dfrac{19 \times 4}{21 \times 4} = \dfrac{76}{84} \\[1em] \Rightarrow \dfrac{23 \times 3}{28 \times 3} = \dfrac{69}{84} \\[1em] \Rightarrow \dfrac{78}{84} \gt \dfrac{76}{84} \gt \dfrac{72}{84} \gt \dfrac{69}{84}$

(13 : 14) > (19 : 21) > (6 : 7) > (23 : 28).

Hence, the ratios in descending order are (13 : 14) > (19 : 21) > (6 : 7) > (23 : 28).

(iii) Given,

(7 : 12), (9 : 16), (13 : 20) and (5 : 8)

We convert them into equivalent like fractions.

L.C.M of 12, 16, 20, 8 is 240.

$\Rightarrow \dfrac{7 \times 20}{12 \times 20} = \dfrac{140}{240} \\[1em] \Rightarrow \dfrac{9 \times 15}{16 \times 15} = \dfrac{135}{240} \\[1em] \Rightarrow \dfrac{13 \times 12}{20 \times 12} = \dfrac{156}{240} \\[1em] \Rightarrow \dfrac{5 \times 30}{8 \times 30} = \dfrac{150}{240} \\[1em] \Rightarrow \dfrac{156}{240} \gt \dfrac{150}{240} \gt \dfrac{140}{240} \gt \dfrac{135}{240}$

(13 : 20) > (5 : 8) > (7 : 12) > (9 : 16).

Hence, the ratios in descending order are (13 : 20) > (5 : 8) > (7 : 12) > (9 : 16).