Question

What is the mean proportion of \[25:10::10:4.\]
(a) 25
(b)10
(c) 4
(d) 100

Answer 1

Hint: In this question, we will use the formula from definition of mean proportion to find the value of mean proportion of given question.The mean proportion can be only calculated if term b=c i.e means should be the same.

Complete step-by-step answer:
If the ratio of two numbers, let say $a$ and $b$ is equal to the ratio of other two numbers, let say $c$ and $d$, then they are said to be proportional.
That is, if $\dfrac{a}{b}=\dfrac{c}{d}$, then both the ratios $a:b$ and $c:d$ are said to be proportional. We denote proportionality by the symbol $::$. Hence, we write, $a:b::c:d$.
Now, if we have b=c, that is, $a:b::b:d$ then we define a term mean proportion. Mean proportion of $a$ and $d$ is defined as the square root of the product of $a$ and $d$ .
That is, mean proportion for $a:b::b:d$ is,
$\sqrt{ad}$.
Again, if we solve the proportionality $a:b::b:d$ for value of b, we get,
$\dfrac{a}{b}=\dfrac{b}{d}$
Cross-multiplying, we get,
$ad={{b}^{2}}$
Taking root on both sides we get,
$b=\sqrt{ad}$
Hence, we see that mean proportion is the value of $b$ in the proportionality $a:b::b:d$.
Therefore, for \[25:10::10:4\], where $a=25,\,b=10,\,d=4$
Mean proportion is,
$\begin{align}
  & \sqrt{ad}=\sqrt{25\times 4} \\
 & =\sqrt{100} \\
\end{align}$
Which is equal to$b$.
Hence the correct answer is option (b).

Note: In this question, if the concept of mean proportion is clear to you, then you can directly, without doing calculation, say that answer will be equal to 10, which is the term $b=c$.