Question

Find the mean proportional between:

(i) 6 and 24
(ii) 3 and 27
(iii) 0.4 and 0.9

Answer 1

(i) Suppose that x is the mean proportional.

Then, 6 : x :: x : 24

⇒ $6\times 24=x\times x$ (Product of extremes = Product of means)
⇒ $x^2=144$
⇒ x = 12

Hence, the mean proportional to 6 and 24 is 12.

(ii) Suppose that x is the mean proportional.

Then, 3 : x :: x : 27
$$\begin{gathered} \Rightarrow 3\times 27=x\times x \\ \Rightarrow x^2=81 \end{gathered}$$ (Product of extremes =Product of means)
⇒ x = 9

Hence, the mean proportional to 3 and 27 is 9.

(iii) Suppose that x is the mean proportional.

Then, 0.4 : x :: x : 0.9

$$\begin{gathered} \Rightarrow 0.4\times 0.9=x\times x \\ \Rightarrow x^2=0.36 \end{gathered}$$ (Product of extremes =Product of means)
⇒x = 0.6

Hence, the mean proportional to 0.4 and 0.9 is 0.6.