Question

Find the mean proportional for each of the following:

(ii) $\frac{1}{16},\frac{4}{25}$

Answer 1

Calculate the mean proportional:

Let $x$ be the mean proportional.

If $x$ is the mean proportional between $a$ and $b$, then $a:x::x:b$.

Then for the numbers $\frac{1}{16},\frac{4}{25}$, the equivalent proportional expression is $\frac{1}{16}:x::x:\frac{4}{25}$.

$$\begin{gathered} \Rightarrow \frac{1}{16}:x=x:\frac{4}{25}(Changingtheproportionalitysigntoequalsign) \\ \Rightarrow \frac{{\displaystyle \frac{1}{16}}}{x}=\frac{x}{{\displaystyle \frac{4}{25}}} \\ \Rightarrow \frac{1}{16}·\frac{4}{25}=x·x(Crossmultiplyingbothsides) \\ \Rightarrow \frac{1}{100}=x^2 \\ \therefore \pm \frac{1}{10}=x(Takingsquarerootonbothsides) \end{gathered}$$

As $-\frac{1}{10}$ does not lie between the given two numbers, therefore, the mean proportional is $\frac{1}{10}$.

Final Answer: The mean proportional is $\frac{1}{10}$.