Compare the following ratios :

(i) (7 : 9) and (11 : 16)

(ii) (19 : 25) and (17 : 20)

(iii) $\Big(\dfrac{1}{2} : \dfrac{1}{5}\Big)$ and (5 : 2)

(i) To compare 2 ratios, the consequent of the first ratio and 2nd ratio must be made equal.

Given,

A : B = 7 : 9 and C : D = 11: 16

L.C.M. of 9 and 16 is 144.

$\Rightarrow \dfrac{A}{B} = \dfrac{7 \times 16}{9 \times 16} = \dfrac{112}{144} \\[1em] \Rightarrow \dfrac{C}{D} = \dfrac{11 \times 9}{16 \times 9} = \dfrac{99}{144} \\[1em] \Rightarrow \dfrac{112}{144} \gt \dfrac{99}{144} \\[1em] \Rightarrow \dfrac{7}{9} \gt \dfrac{11}{16}$

Hence, 7 : 9 > 11 : 16.

(ii) To compare 2 ratios, the consequent of the first ratio and 2nd ratio must be made equal.

Let A : B = 19 : 25 and C : D = 17 : 20.

L.C.M. of 25 and 20 is 100.

$\Rightarrow \dfrac{A}{B} = \dfrac{19 \times 4}{25 \times 4} = \dfrac{76}{100} \\[1em] \Rightarrow \dfrac{C}{D} = \dfrac{17 \times 5}{20 \times 5} = \dfrac{85}{100} \\[1em] \Rightarrow \dfrac{76}{100} \lt \dfrac{85}{100} \\[1em] \Rightarrow \dfrac{19}{25} \lt \dfrac{17}{20}$

Hence, 19 : 25 < 17 : 20.

(ii) To compare 2 ratios, the consequent of the first ratio and 2nd ratio must be made equal.

Let A : B = $\Big(\dfrac{1}{2} : \dfrac{1}{5}\Big)$ and C : D = 5:2

First simplifying A : B,

L.C.M of 2 and 5 is 10.

$\Rightarrow \Big(\dfrac{1}{2} \times 10 : \dfrac{1}{5} \times 10\Big) \\[1em] \Rightarrow 5:2$

Since both ratios are same.

Hence, $\Big(\dfrac{1}{2} : \dfrac{1}{5}\Big)$ = 5 : 2.