Find the third proportional to :

(i) 36, 12

(ii) 1.2, 0.6

(iii) $\dfrac{1}{9}, \dfrac{2}{3}$

(iv) 1m 60 cm, 40 cm

(v) 1 kg 250 g, 500 g

(vi) ₹ 2.40, ₹ 4.80

(i) 36, 12

Let the third proportional be x.

Then, 36 : 12 :: 12 : x.

product of extremes = product of means

36 × x = 12 × 12

⇒ x = $\dfrac{144}{36}$

⇒ x = 4

Hence, the third proportional is 4

(ii) 1.2, 0.6

Let the third proportional be x.

Then, 1.2 : 0.6 :: 0.6 : x.

product of extremes = product of means

1.2 × x = 0.6 × 0.6

⇒ x = $\dfrac{0.36}{1.2}$

⇒ x = $\dfrac{3.6}{12}$

⇒ x = 0.3

Hence, the third proportional is 0.3

(iii) $\dfrac{1}{9}, \dfrac{2}{3}$

Let the third proportional be x.

Then, $\dfrac{1}{9} : \dfrac{2}{3} :: \dfrac{2}{3} : x$.

product of extremes = product of means

$\dfrac{1}{9} \times x = \dfrac{2}{3} \times \dfrac{2}{3} \\[1em] \Rightarrow \dfrac{x}{9} = \dfrac{4}{9} \\[1em] \Rightarrow x = \dfrac{4}{9} \times 9 \\[1em] \Rightarrow x = 4$

Hence, the third proportional is 4

(iv) 1 m 60 cm, 40 cm

First, convert to the same unit:

1 m = 100 cm

∴ 1 m 60 cm = 100 cm + 60 cm = 160 cm.

Let the third proportional be x.

Then, 160 : 40 :: 40 : x

product of extremes = product of means

160 × x = 40 × 40

⇒ x = $\dfrac{1600}{160}$

⇒ x = 10

Hence, the third proportional is 10 cm

(v) 1 kg 250 g, 500 g

First, convert to the same unit:

1 kg = 1000 g

∴ 1 kg 250 g = 1000 g + 250 g = 1250 g

Let the third proportional be x.

Then, 1250 : 500 :: 500 : x

product of extremes = product of means

1250 x x = 500 x 500

⇒ x = $\dfrac{250000}{1250}$

⇒ x = $\dfrac{25000}{125}$

⇒ x = 200

Hence, the third proportional is 200 g

(vi) ₹ 2.40, ₹ 4.80

Let the third proportional be x.

Then, 2.40 : 4.80 :: 4.80 : x.

product of extremes = product of means

2.40 × x = 4.80 × 4.80

⇒ x = $\dfrac{4.80 \times 4.80}{2.40}$

⇒ x = 2 x 4.80

⇒ x = 9.60

Hence, the third proportional is ₹ 9.60