Question

Find the mean proportional between:
$\left(i\right)$ $6$ and $24$
$\left(ii\right)$ $3$ and $27$
$\left(iii\right)$ $0.4$ and $0.9$

Answer 1

$\left(i\right)$ $6$ and $24$

Suppose that $x$ is the mean proportional of $6$ and $24$
Then, $6:x::x:24$

$\Rightarrow 6\times 24=x\times x$ [ Since, Product of extremes = Product of means]

$\Rightarrow 144=x^2$

$\Rightarrow \sqrt{144}=x$

$\Rightarrow 12=x$

Hence, the mean proportional between $6$ and $24$ is $12$.
$\left(ii\right)$ $3$ and $27$

Suppose that $x$ is the mean proportional of $3$ and $27$

Then, $3:x::x:27$

$\Rightarrow 3\times 27=x\times x$ [ Since, Product of extremes = Product of means]

$\Rightarrow 81=x^2$

$\Rightarrow \sqrt{81}=x$

$\Rightarrow 9=x$

Hence, the mean proportional between $3$ and $27$ is $9$.

$\left(iii\right)$ $0.4$ and $0.9$

Suppose that $x$ is the mean proportional of $0.4$ and $0.9$

Then, $0.4:x::x:0.9$

$\Rightarrow 0.4\times 0.9=x\times x$ [ Since, Product of extremes = Product of means]

$\Rightarrow 0.36=x^2$

$\Rightarrow \sqrt{0.36}=x$

$\Rightarrow 0.6=x$

Hence, the mean proportional between $0.4$ and $0.9$ is $0.6$.

Alternate method:

Mean proportional between $a$ and $b$ is $\sqrt{ab}$.

$\left(i\right)$ Mean proportional between $6$ and $24$ $=\sqrt{6\times 24}=\sqrt{144}=12$

$\left(ii\right)$ Mean proportional between $3$ and $27$ $=\sqrt{3\times 27}=\sqrt{81}=9$

$\left(iii\right)$ Mean proportional between $0.4$ and $0.9$ $=\sqrt{0.4\times 0.9}=\sqrt{0.36}=0.6$.