Question

The mean proportion between $5:45$ is
$A) 25$
$B) 15$
$C) 5$
$D) 115$

Answer 1

Hint: Here we have to find the mean proportion between two numbers. Using the mean proportional property, to write it’s in the ratio form and then we have to do some simplification. Finally we get the required answer.

Complete step by step solution:
It is given that the two numbers are $5$ and $45$
We have to find out the Mean proportion of the given two numbers.
We know the relation of the mean proportional between the two numbers $(a,b)$ is given as $a:x::x:b$, where $x$ is the mean proportional,
Let the mean proportion of these numbers be $x$.
Then according to mean proportional property convert then in equation which is given as:
$5:x::x:45$
We are just writing the expression in the form of fraction because it is in ratio,
$ \Rightarrow \dfrac{5}{x} = \dfrac{x}{{45}}$
Now we need to cross multiply the two fractions and multiply the numerator of the first fraction to the denominator of the second fraction.
So we have to equating the both sides and we can write it as,
$ \Rightarrow x \times x = 5 \times 45$
We multiply the terms in each side
$ \Rightarrow {x^2} = 225$
Taking square on both sides we get,
$ \Rightarrow x = \sqrt {225} $
On split the root value as,
$ \Rightarrow x = \sqrt {15 \times 15} $
We can write it as,
$ \Rightarrow x = 15$
So, the mean proportion of $5$ and $45$ is $15$.
Thus, we get the required answer.

Hence the correct option is $\left( b \right).$

Note: In this particular type of the question it is important to note that the term “mean proportion” may also be referred to as “geometric mean”. Mean proportional, or geometric mean, is not the same as the arithmetic mean. While an arithmetic mean deals with addition, a geometric mean deals with multiplication.