Question

Find the mean proportional between $3$ and $12$.

Answer 1

Continued proportion:

If $\mathrm{a},\mathrm{b}$ and$\mathrm{c}$ are in continued proportion, then $\mathrm{b}$ is called the mean proportionalof $\mathrm{a}$ and $\mathrm{c}$.

$$\begin{gathered} \mathrm{a}:\mathrm{b}=\mathrm{b}:\mathrm{c} \\ \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{b}}{\mathrm{c}} \\ {\mathrm{b}}^2=\mathrm{a}\times \mathrm{c} \end{gathered}$$

Let $\mathrm{a}=3,\mathrm{c}=12$, then

$\begin{array}{rcl}{b}^2& =& a\times c\\ & =& 3\times 12\\ \mathrm{b}& =& \sqrt{36}\\ b& =& 6\end{array}$

Hence, mean proportional between $3$ and $12$ is $6$.