Question

How do you convert $0.2$ ( $2$ being repeated) to a fraction?

Answer 1

Hint: Firstly we should assume a variable $x$ and equate it to the given number. Since $2$ is recurring we can write the given decimal number as $0.22$. Now we will multiply both sides by $10$. After that, we can subtract both sides by their initial equation. Post which we can keep the variables on one side and constants on the other. This would be the last step as we will be getting the required fraction.

Complete step by step solution:
First step is to assume the given decimal is equal to a variable $x$.
$\Rightarrow x = 0.2............(1)$
Since $2$is recurring we can write the given decimal number as$0.22$
\[\Rightarrow x = 0.22............(2)\]
Multiplying the above equation by$10$ we get the following equation
$\Rightarrow 10x = 2.2............(3)$
Subtracting equation $1$ from equation $3$
$\Rightarrow 10x - x = 2.2 - 0.2............(4)$
$\Rightarrow 9x = 2.0............(5)$
Moving all the variables and all the constants on one side we get the required fraction.
$\Rightarrow x = \dfrac{2}{9}$.

Note: This method is very simple even in the case where the recurring digit is at $1000^{th}$ decimal place, for example, $0.23444$, for this the recurring digit is $4$. For this particular sum, the method is the same as above where we first multiply by $10$ and then equate the variable. Students should always follow this method as it does not involve any effort in memorizing the complex fraction, nor they have to think about the divisor which would bring the given decimal in the fraction form.