Question

Find two numbers such that the mean proportion between them is 4 and the third
proportion is 32.

Answer 1

Answer :

2 and 8

Explanation :

Assuming the two numbers as $\boldsymbol{x \: \& \: y}$

${ \therefore \sf \: Mean \: proportion = \boldsymbol{x :4 : : 4 : y}}$

$\implies \boldsymbol{ \dfrac{x}{4} = \dfrac{4}{y} }$

$\implies \boldsymbol{xy = 4 \times 4}$

$\implies \boldsymbol{xy = 16}$

$\implies \boldsymbol{x = \dfrac{16}{y}. . . . . .(i)}$

And also,

Third proportion given is 32.

Then,

$\implies \boldsymbol{x : y : : y : 32 }$

$\implies \boldsymbol{ \dfrac{x}{y} = \dfrac{y}{32} }$

$\implies \boldsymbol{32x = {y}^{2} . . . . . .(ii) }$

From eq. (i)

$\implies \boldsymbol{32 \times \dfrac{16}{y} = {y}^{2} }$

$\implies \boldsymbol{32 \times 16 = {y}^{3} }$

$\implies \boldsymbol{512 = {y}^{3} }$

$\implies \boldsymbol{\sqrt[3]{512} = y}$

$\therefore \boldsymbol{y = 8}$

Substitute the value of $\boldsymbol{y}$ in eq. (ii) we get,

$\implies \boldsymbol{32x = {y}^{2} }$

$\implies \boldsymbol{32x = {(8)}^{2} }$

$\implies \boldsymbol{32x =64 }$

$\implies \boldsymbol{x = \dfrac{64}{32} }$

$\therefore \boldsymbol{x = 2}$

Hence, the two numbers are 2 and 8