Question

Find the mean proportion to 0.3 and 2.7.

Answer 1

Hint: Mean proportion of two numbers is the square root of the product of these numbers. To find the mean proportion to 0.3 and 2.7, we have to take the square root of the product of 0.3 and 2.7.

Complete step by step answer:
We have to find the mean proportion to 0.3 and 2.7. We know that the mean proportion of two numbers is the square root of the product of these numbers. Let us consider two numbers a and b. The mean proportion of a and b is given by
$\sqrt{a\times b}$ .
Let us find the mean proportion to 0.3 and 2.7.
$\Rightarrow \text{Mean proportion}=\sqrt{0.3\times 2.7}$
We have to multiply 0.3 and 2.7.
$\Rightarrow \text{Mean proportion}=\sqrt{0.81}$
We know that 81 is a perfect square, that is, $9\times 9=81$ . Therefore, we can write 0.81 as the product of 0.9 and 0.9.
$\begin{align}
  & \Rightarrow \text{Mean proportion}=\sqrt{0.9\times 0.9} \\
 & \Rightarrow \text{Mean proportion}=\sqrt{{{0.9}^{2}}} \\
\end{align}$
We know that $\sqrt[n]{{{a}^{n}}}=a$ . Therefore, we can write the above result as
$\Rightarrow \text{Mean proportion}=0.9$

Hence, the mean proportion to 0.3 and 2.7 is 0.9.

Note: Mean proportion is also known as Geometric mean. Students may get confused with geometric mean and arithmetic mean. Arithmetic mean is the sum of all of the numbers of a group divided by the number of items in that list. We can also find the mean proportion to 0.3 and 2.7 by writing the proportion in the following manner.
$\Rightarrow 0.3:x::x:2.7$
We know that $a:b::c:d$ means $\dfrac{a}{b}=\dfrac{c}{d}$ . Therefore, we can write the above form as
$\Rightarrow \dfrac{0.3}{x}=\dfrac{x}{2.7}$
Let us cross multiply.
$\begin{align}
  & \Rightarrow {{x}^{2}}=0.3\times 2.7 \\
 & \Rightarrow {{x}^{2}}=0.81 \\
\end{align}$
We have to take square root of both the sides.
$\begin{align}
  & \Rightarrow x=\sqrt{0.81} \\
 & \Rightarrow x=0.9 \\
\end{align}$
Hence, the mean proportion to 0.3 and 2.7 is 0.9.