Question

Express \[0.\overline {23} \] as a rational number in simplest form?

Answer 1

Hint: To convert a decimal to fraction, first divide decimal by 1 to make given decimal to fraction, then multiply both numerator and denominator by 10 to make a decimal to whole number in number and further simplify by using highest common factor of both numerator and denominator.

Complete step by step solution:
To find the equivalent fraction for decimal point number \[0.\overline {23} \] manually, The bar represents the number that is repeating.
Consider the given decimal value
\[ \Rightarrow \,\,x = \,0.23\]
To write it as a fraction by divide 1
\[ \Rightarrow \,\,\,\,\dfrac{{0.23}}{1}\]
To make decimal to whole number in numerator
Multiply both numerator and denominator by 100 for every number after the decimal point:
As we have two numbers after the decimal point, hence we multiply both numerator and denominator by 100. then,
\[ \Rightarrow \,\,\,\,\dfrac{{0.23}}{1} \times \dfrac{{100}}{{100}}\]
\[ \Rightarrow \,\,\,\,\dfrac{{23.23}}{{100}}\]
since 23 being repeated so number is written as above
\[ \Rightarrow \,\,x = \,\,\dfrac{{23.23}}{{100}}\]
Take 100 to LHS
\[ \Rightarrow \,\,100x = \,\,23.23\]
Add -x on both sides we have
\[ \Rightarrow \,\,100x - x = \,\,23.23 - x\]
As we know that the value of $x$ is $0.23$, by substituting we have
\[ \Rightarrow \,\,99x = \,\,23.23 - 0.23\]
\[ \Rightarrow \,\,99x = \,\,23\]
Divide the above equation by 99 we get
\[ \Rightarrow \,\,x = \,\dfrac{{23}}{{99}}\]
Therefore, the simplest form of \[0.\overline {23} \] is $\dfrac{23}{99}$.
Hence, we have converted the decimal into a fraction. The obtained fraction is an improper fraction.

Note:
The number can be converted from one form to the other form. For the conversion of numbers, we have some rules or methods. By using the specific methods and rules we can convert the number. So to convert the decimal number we have to multiply and divide the number by multiples of 10. The multiplication of 10 depends on the number present after the decimal point.