Find the fourth proportional to :

(i) 4, 9, 32

(ii) 15, 6, 7

(iii) 0.6, 1.5, 3

(iv) $\dfrac{1}{3}, \dfrac{2}{5}, 6$

(v) $2\dfrac{1}{2}, 2\dfrac{6}{7}, 3\dfrac{1}{2}$

(vi) 3 hrs 12 min, 24 min, 1 m 68 cm

(i) 4, 9, 32

Let the fourth proportional be x. Then 4, 9, 32, x are in proportion.

product of extremes = product of means

4 × x = 9 × 32

⇒ x = $\dfrac{9 \times 32}{4}$

⇒ x = 9 × 8 = 72

Hence, the fourth proportional is 72.

(ii) 15, 6, 7

Let the fourth proportional be x. Then 15, 6, 7, x are in proportion.

product of extremes = product of means

15 × x = 6 × 7

⇒ x = $\dfrac{42}{15} = \dfrac{14}{5} = 2.8$

Hence, the fourth proportional is 2.8

(iii) 0.6, 1.5, 3

Let the fourth proportional be x. Then 0.6, 1.5, 3, x are in proportion.

product of extremes = product of means

0.6 × x = 1.5 × 3

⇒ x = $\dfrac{4.5}{0.6} = \dfrac{4.5 \times 10}{0.6 \times 10} = \dfrac{45}{6} = 7.5$

Hence, the fourth proportional is 7.5

(iv) $\dfrac{1}{3}, \dfrac{2}{5}, 6$

Let the fourth proportional be x. Then $\dfrac{1}{3}, \dfrac{2}{5}, 6$, x are in proportion.

product of extremes = product of means

$\dfrac{1}{3} \times x = \dfrac{2}{5} \times 6 \\[1em] \Rightarrow \dfrac{x}{3} = \dfrac{12}{5} \\[1em] \Rightarrow x = \dfrac{12 \times 3}{5} \\[1em] \Rightarrow x = \dfrac{36}{5} \\[1em] \Rightarrow x = 7\dfrac{1}{5}$

Hence, the fourth proportional is $7\dfrac{1}{5}$

(v) $2\dfrac{1}{2}, 2\dfrac{6}{7}, 3\dfrac{1}{2}$

Convert to improper fractions: $\dfrac{5}{2}, \dfrac{20}{7}, \dfrac{7}{2}$. Let the fourth proportional be x.

Then $\dfrac{5}{2}, \dfrac{20}{7}, \dfrac{7}{2}$, x are in proportion.

product of extremes = product of means

$\dfrac{5}{2} \times x = \dfrac{20}{7} \times \dfrac{7}{2} \\[1em] \Rightarrow \dfrac{5x}{2} = 10 \\[1em] \Rightarrow 5x = 10 \times 2 \\[1em] \Rightarrow 5x = 20 \\[1em] \Rightarrow x = \dfrac{20}{5} \\[1em] \Rightarrow x = 4$

Hence, the fourth proportional is 4

(vi) 3 hrs 12 min, 24 min, 1 m 68 cm

First, convert to the same units for each ratio:

1 hour = 60 min,

∴ 3 hrs = 3 x 60 min = 180 min

3 hrs 12 min = 180 min + 12 min = 192 min.

1 m = 100 cm,

∴ 1 m 68 cm = 100 cm + 68 cm = 168 cm.

Let the fourth proportional be x (in cm). We have:

192 : 24 :: 168 : x

Then 192 min, 24 min, 168 cm, x are in proportion.

product of extremes = product of means

192 x x = 24 x 168

⇒ x = $\dfrac{24 \times 168}{192}$

⇒ x = $\dfrac{1 \times 168}{8}$

⇒ x = 21

Hence, the fourth proportional is 21 cm.