Question

How do we convert $0.23$ (3 repeating) to a fraction?

Answer 1

Hint: According to the question, to convert the decimal number into the fraction, firstly we have to count the number of digits after the decimal point. And, then proceed to remove the decimal into fraction.

Complete step-by-step answer:
First we will write the steps to convert repeating decimals to the fraction, the steps are as follows:-
Step-1: Let $x$ equal the repeating decimal we are trying to convert to a fraction.
Step-2: Examine the repeating decimal to find the repeating digit(s).
Step-3: Place the repeating digit(s) to the left of the decimal point.
Step-4: Place the repeating digit(s) to the right of the decimal point.
Step-5: Using the two equations you found in step 3 and step 4, subtract the left sides of the two equations. Then, subtract the right sides of the two equations: as we subtract, just make sure that the difference is positive for both sides.
Now, we will solve the given repeating decimal on the basis of the above steps:-
Step-1: Let $$x=0.23333333333$$
Step-2: After examination, the repeating digit is 3.
Step-3: To place the repeating digit (3) to the left of the decimal point, we need to move the decimal point 1 place to the right.
Technically, moving a decimal point one place to the right is done by multiplying the decimal number by 10.
When we multiply one side by a number, we have to multiply the other side by the same number to keep the equation balanced.
Thus, $10x=2.33333333333$
Step-4: So, we have two equations now,
 $10x=2.33333333333$
 $$x=0.23333333333$$
Now, the difference of both the equation are:
 $$\begin{gathered} 10x-x=2.333333333-0.233333333 \\ \Rightarrow 9x=2.1 \end{gathered}$$
Now, to remove the decimal divide by 10 because there is no any repeating digit(s)
 $$\begin{gathered} \Rightarrow 9x={\displaystyle \frac{21}{10}} \\ \Rightarrow x={\displaystyle \frac{21}{10\times 9}}={\displaystyle \frac{21}{90}}={\displaystyle \frac{7}{30}} \end{gathered}$$
Hence, the fraction of $0.23$ (3 repeating) is $${\displaystyle \frac{7}{30}}$$.

So, the correct answer is “ $${\displaystyle \frac{7}{30}}$$ ”.

Note: In fraction number, when the numerator is greater than the denominator then the fraction is called Improper Fraction and we can convert it into mixed fraction by dividing the numerator from denominator.