Question

The value of $ 5.\overline 2 $ is equal to
A. $ \dfrac{{45}}{9} $
B. $ \dfrac{{46}}{9} $
C. $ \dfrac{{47}}{9} $
D. None of these

Answer 1

Hint: In this problem, we need to find a rational number whose decimal expansion is non-terminating and repeating. First we will assume the given number as $ x $ and then we will multiply it by $ 10 $ . After simplification, we will get the required answer.

Complete step-by-step answer:
In this problem, we need to find the value of rational number $ 5.\overline 2 $ whose decimal expansion is non-terminating and repeating. We know that $ 5.\overline 2 $ can be written as $ 5.222222 \ldots $ .
Now we will find the value of $ 5.222222 \ldots $ . For this, let us assume that $ x = 5.\overline 2 = 5.222222 \ldots \cdots \cdots \left( 1 \right) $ . Here we can see that only one digit is repeated. So, we will multiply by $ 10 $ on both sides of equation $ \left( 1 \right) $ . Therefore, we get
 $
  10x = 10 \times \left( {5.222222 \ldots } \right) \\
   \Rightarrow 10x = 52.222222 \ldots \cdots \cdots \left( 2 \right) \\
  $
Let us subtract equation $ \left( 1 \right) $ from equation $ \left( 2 \right) $ . Therefore, we get
 $
  10x - x = \left( {52.222222 \ldots } \right) - \left( {5.222222 \ldots } \right) \\
   \Rightarrow 9x = 47 \cdots \cdots \left( 3 \right) \\
  $
Now we can say that the equation $ \left( 3 \right) $ is a linear equation in one variable $ x $ . Let us solve the equation $ \left( 3 \right) $ to find $ x $ . Let us divide by $ 9 $ on both sides of the equation $ \left( 3 \right) $ . So, we get
 $
  \dfrac{{9x}}{9} = \dfrac{{47}}{9} \\
   \Rightarrow x = \dfrac{{47}}{9} \\
   \Rightarrow 5.\overline 2 = \dfrac{{47}}{9} \\
  $
Therefore, we can say that the value of $ 5.\overline 2 $ is equal to $ \dfrac{{47}}{9} $ . Therefore,
So, the correct answer is “Option C”.

Note: If a digit or a sequence of digits is repeated infinitely in a decimal then that decimal is called non-terminating repeating or recurring decimal. The recurring decimal can be written by putting a bar over the repeated digit. If we want to find the value of $ 5.\overline {24} $ then first we will write $ 5.\overline {24} $ as $ 5.242424 \ldots $ . Here we can see that two digits are repeated. Therefore, we will multiply by $ 100 $ on both sides of equation $ x = 5.242424 \ldots $ . Then, we will apply the same procedure as we discussed in the given problem.