Question

Convert $0.36\left( {6\,{\text{repeating}}} \right)$ to a fraction.

Answer 1

Hint: To convert the given recurring decimal into fraction we have to first eliminate the recurring part of the decimal. For that we have to form two equations such that while subtracting those two equations we can eliminate the recurring part. Also on further simplification of those two equations, we can thus find the required fraction.

Complete step by step answer:
Given
$0.366666..\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.....................\left( i \right)$
Now we need to convert $0.366666..$ into a fraction, for that we have to form two equations such that while subtracting those two equations we can eliminate the recurring part.
So forming two equations:
Let \[x = 0.366666..\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,................\left( {ii} \right)\]
Now for two equations let us consider $10x\,\,{\text{and}}\,\,100x$ such that:
$10x = 3.66666..\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,..............\left( {iii} \right)$
$100x = 36.6666..\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,..............\left( {iv} \right)$
So our required equations are (iii) and (iv).
Now subtract (iii) from (iv), such that:
$
  100x - 10x = (36.6666…….)\, - (3.6666………) \\
 90x = 33.............................\left( v \right) \\
 $
On observing (v) we can say that we have eliminated the recurring part, now we just have to simplify the equation (v) to get our required fraction.
So simplifying (v), we get:
\[
\Rightarrow 90x = 33 \\
\Rightarrow x = \dfrac{{33}}{{90}} \\
\Rightarrow x = \dfrac{{3 \times 11}}{{3 \times 30}} \\
\Rightarrow x = \dfrac{{11}}{{30}} \\
 \]
From (i) we can write:
\[ \Rightarrow x = (0.366666……….) = \dfrac{{11}}{{30}}...........\left( {vi} \right)\]
So from (vi) we can write that on converting $0.366666..$ to fraction we get \[\dfrac{{11}}{{30}}.\]
Therefore our required answer is \[\dfrac{{11}}{{30}}\].

Note: In order to solve similar questions some standard steps can be followed as mentioned below:
1. Consider x to be the given recurring decimal.
2. Let n be the number of recurring digits
3. Now multiply the recurring decimal with multiples of ${10^n}$.
4. Subtract the two equations formed by us to eliminate the recurring part.
5. Now simplify and solve for x to find the required fraction.
The above steps can be performed to easily obtain the fraction from recurring decimal numbers. Also care must be taken while subtracting since the subtraction involving decimals can be tricky.