Question

Find the mean proportion between \[4\] and \[25\].

Answer 1

Hint: To solve this question first we assume the mean proportion. Then apply the condition of that mean proportion and try to find the value of that mean proportion.
The mean proportion is also known as the geometric proportion. The mean proportion of two numbers \[a\] and \[b\] is \[r\] then the relation between \[a,b,r\] is \[{r^2} = ab\]

Complete step-by-step answer:
Given, Two numbers are given \[4\] and \[25\].
We need to find the mean proportion between \[4\] and \[25\].
The mean proportion is also known as the geometric proportion. The mean proportion of two numbers \[a\] and \[b\] is \[r\] then the relation between \[a,b,r\] is \[{r^2} = ab\]
The relation comes from the equation of geometric proportion.
\[\dfrac{a}{r} = \dfrac{r}{b}\]
On rearranging we get the relation
\[{r^2} = ab\]
Here, value of \[a\] is 4 and
Value of \[b\] is 25
On putting the values in the relation.
\[{r^2} = ab\]
\[{r^2} = 4 \times 25\]
On further calculating
\[{r^2} = 100\]
Taking root both side
\[\sqrt {{r^2}} = \sqrt {100} \]
From here value of r is
\[r = \pm 10\]
Here we take only positive value because the mean proportion of any two positive numbers is never negative.
The value of the mean proportion between \[4\] and \[25\] is \[r = 10\].

Note: To solve this type of question we have to first determine the value of $a$ and $b$ the put all that in the relation of mean proportion or geometric proportion of that numbers and always take the sign according to the question because they might be negative or positive because we are unable to find the root of the negative number. And the root of a positive number is positive and negative numbers. A negative is always applicable if both the numbers are negative.