Question

Find the approximate value of \[\dfrac{{1 + \sqrt {0.01} }}{{1 - \sqrt {0.1} }}\].
A.0.6
B.1.1
C.1.6
D.1.7

Answer 1

Hint: Here in this question, we have to find the approximate value of a given fraction. The fraction having a radical first we need to determine the value of radicals or square root of a number by the division or any other method. And further simplify by using a basic arithmetic operation we get the required solution.

Complete step-by-step answer:
Consider the given fraction:
\[\dfrac{{1 + \sqrt {0.01} }}{{1 - \sqrt {0.1} }}\] --------(1)
We need to find it’s approximate value
Here, the fraction having a radical or square root number in both numerator and denominator. First, we need to find that value.
Take, \[\sqrt {0.01} \]
It can be written as
\[ \Rightarrow \,\,\,\sqrt {\dfrac{1}{{100}}} \]
By quotient rule of property, then
\[ \Rightarrow \,\,\,\dfrac{{\sqrt 1 }}{{\sqrt {100} }}\]
As we know 1 itself a square number and 100 is a square number of 10, then
\[ \Rightarrow \,\,\,\dfrac{1}{{10}}\]
On simplification we get
\[ \Rightarrow \,\,\,0.1\]
\[\therefore \] The value of \[\sqrt {0.01} = 0.1\]-----(2)
Now, find the value of \[\sqrt {0.1} \] by division method, we have
\[\begin{align}
 & {0.316 \\ 3}\left| \!{\overline {\,
 \begin{align}
&\overline{00}. \ \overline{10} \ \overline{00} \ \overline{00} \\
 & \underline{009} \\
 & 1 \\
\end{align} \,}} \right. \\
 & 61\left| \!{\overline {\,
 \begin{align}
  & 1\overline{00} \\
 & \underline{61} \\
 & 39 \\
\end{align} \,}} \right. \\
 & 626\left| \!{\overline {\,
 \begin{align}
  & 39\overline{00} \\
 & \underline{3756} \\
 & 144 \\
\end{align} \,}} \right. \\
\end{align}\]
\[\therefore \] the value of \[\sqrt {0.1} = 0.316\]----(3)
Now substitute the values of (2) and (3) in fraction (1)
Consider,
\[ \Rightarrow \dfrac{{1 + \sqrt {0.01} }}{{1 - \sqrt {0.1} }}\]
\[ \Rightarrow \dfrac{{1 + 0.1}}{{1 - 0.316}}\]
On simplification, we get
\[ \Rightarrow \dfrac{{1.1}}{{0.684}}\]
On division, we get
\[ \Rightarrow 1.60818\]
Hence, the approximate value of \[\dfrac{{1 + \sqrt {0.01} }}{{1 - \sqrt {0.1} }} \simeq 1.6\].
Therefore, option (C) is correct.
So, the correct answer is “Option C”.

Note: While simplifying the fraction we have to divide both the numerator and denominator by a common number and while finding a square root value should know the division method is a specified method. Remember the basic arithmetic operations like addition, subtraction, multiplication and division to simplify the fractions and rounding off number is a major role in finding an approximate value.