Question

How do you convert 0.23 (3 being repeated) to a fraction.

Answer 1

Hint:
 We will recall the concept of fraction and conversion of decimal to fraction to solve the above question. We can represent 0.23 (3 being repeated) as $ 0.2\bar{3} $ , where bar at the top symbolises that 3. So, let us say that $ x=0.2\bar{3} $ , then $ 10x=2.\bar{3}=2.3\bar{3} $ , and $ 100x=23.\bar{3} $ . Then, we will subtract 100x with 10x so that all the 3 which is repeating after decimal get cancelled. Then, we will solve the obtained equation $ \left( 100x-10x \right)=23.\bar{3}-2.\bar{3} $ to get the fractional value of x.

Complete step by step answer:
We will use the concept of the linear equation, fraction, and conversion of decimal to fraction to solve the above question.
We can see from the question that in 0.23, 3 is being repeated. So, we can write it as $ 0.2\bar{3} $, where bar at the top represents that 3 is repeating and 0.23 is a repeating decimal.
Let us assume that x is equal to $ 0.2\bar{3} $ .
 $ \Rightarrow x=0.2\bar{3}----\left( 1 \right) $
Now, when we multiply both side of the equation (1) by 10, we will get:
 $ \Rightarrow 10x=2.3\bar{3}---\left( 2 \right) $
And, when we multiply both side of the equation by 100, we will get:
 $ \Rightarrow 100x=23.3\bar{3}---\left( 3 \right) $ , because 3 is repeating.
Now, after subtracting equation (2) from (3) we will get:
 $ \Rightarrow \left( 100x-10x \right)=23.3\bar{3}-2.3\bar{3} $
When we subtract $ 2.3\bar{3} $ from $ 23.3\bar{3} $ , all the 3 after decimal gets converted to 0.
 $ \Rightarrow 90x=21 $
 $ \Rightarrow x=\dfrac{21}{90} $
Now, after dividing numerator and denominator both by 3 we will get:
 $ \therefore x=\dfrac{7}{30} $
Hence, 0.23 (3 being repeated) can represented as $ \dfrac{7}{30} $ in fraction.
This is our required solution.

Note:
There are two types of a decimal number, first one is a decimal number which has terminated decimal number, it means the last digit after the decimal point is not repeating and the second one is repeating decimal number, it means the last digit after decimal point goes on and it does not have any endpoint and we represent repeating decimal with the bar at the top of the repeated digit.