Question

Express the number $0.3\overline {178} $ in the form of rational number $\dfrac{a}{b}$ .

Answer 1

Hint: Let, $x = 0.3\overline {178} $ .
Firstly, write $0.3\overline {178} $ as 0.3178178….
Then, multiply the number 0.3178178… by 10 and 10000, individually and find the difference between the equations formed.
Finally, write the number x in the form of $\dfrac{a}{b}$ .

Complete step-by-step answer:
Let, $x = 0.3\overline {178} $ .
Also, we can write $x = 0.3\overline {178} $ as $x = 0.3178178178...$
 $\Rightarrow x = 0.3178178178...$ … (1)
Now, multiplying equation (1) by 10, we get
 $10 \times x = 10 \times 0.3178178178...$
 $\Rightarrow 10x = 3.178178178...$ … (2)
Now, multiplying equation (1) by 10000, we get
 $10000 \times x = 1000 \times 0.3178178178...$
 $\Rightarrow 10000x = 3178.178178178...$ … (3)
We will subtract equation (2) from equation (3) to get a rational number.
 \[
  \Rightarrow 10000x - 10x = 3178.178178178... - 3.178178178... \\
  \Rightarrow 9990x = 3175 \\
  \Rightarrow x = \dfrac{{3175}}{{9990}} \\
  \Rightarrow x = \dfrac{{635}}{{1998}} \\
 \]
Thus, the number $0.3\overline {178} $ can be expressed in the form of a rational number as $\dfrac{{635}}{{1998}}$ .

Note: Recurring decimal is also known as repeating decimal. In a decimal, if a digit or a sequence of digits keeps on repeating itself infinite times, then that decimal number is called recurring decimal number.
We put a bar sign on the top of the digits that keep on repeating themselves.
For example, a recurring decimal number is 3.78696969….
Here, the sequence of the digits 6 and 9 keep repeating infinite times.
So, we can write the number 3.78696969…. as $3.78\overline {69} $ .
The bar sign on the top of a digit or a sequence of digits suggest that the digits are repeated infinite times.