Find the value of x in each of the following :

(i) 42 : 12 : : 7 : x

(ii) 1.8 : x : : 2.4 : 6.0

(iii) 6 : 0.8 : : x : 10

(iv) x : 1.6 : : 2.1 : 8.4

(v) $\dfrac{1}{9}$ : x : : $\dfrac{1}{3} : \dfrac{1}{4}$

(vi) 16 : x : : x : 25

(i) 42 : 12 : : 7 : x

In a proportion, we know that:

product of extremes = product of means

∴ 12 x 7 = 42 x x

⇒ 84 = 42x

⇒ x = $\dfrac{84}{42} = 2$

Hence, x = 2

(ii) 1.8 : x : : 2.4 : 6.0

In a proportion, we know that:

product of extremes = product of means

∴ x x 2.4 = 1.8 x 6.0

⇒ 2.4x = 10.8

⇒ x = $\dfrac{10.8}{2.4} = \dfrac{108}{24} = \dfrac{9}{2} = 4.5$

Hence, x = 4.5

(iii) 6 : 0.8 : : x : 10

In a proportion, we know that:

product of extremes = product of means

∴ 0.8 x x = 6 x 10

⇒ 0.8x = 60

⇒ x = $\dfrac{60}{0.8} = \dfrac{600}{8} = 75$.

Hence, x = 75

(iv) x : 1.6 : : 2.1 : 8.4

In a proportion, we know that:

product of extremes = product of means

∴ 1.6 x 2.1 = x x 8.4

⇒ 3.36 = 8.4x

⇒ x = $\dfrac{3.36}{8.4} = \dfrac{336}{840} = 0.4$

Hence, x = 0.4

(v) $\dfrac{1}{9}$ : x : : $\dfrac{1}{3} : \dfrac{1}{4}$

In a proportion, we know that:

product of extremes = product of means

$∴ x \times \dfrac{1}{3} = \dfrac{1}{9} \times \dfrac{1}{4}$

⇒ $\dfrac{x}{3} = \dfrac{1}{36}$

⇒ x = $\dfrac{3}{36} = \dfrac{1}{12}$.

Hence, x = $\dfrac{1}{12}$

(vi) 16 : x : : x : 25

In a proportion, we know that:

product of extremes = product of means

∴ x x x = 16 x 25

⇒ x² = 400

⇒ x = $\sqrt{400} = 20$

Hence, x = 20