Question

Express \[0.\overline {23} \] in \[\dfrac{p}{q}\] form where \[p\] and \[q\] are integers and \[q \ne 0\].

Answer 1

Hint: Here we will assume \[x\] to be \[0.23232323\]. We will then multiply the expression by 100 to obtain the second equation. Then subtract the two equations such that the numbers in the decimal places will be 0. We will further simplify the equation to find the value of \[x\], and hence, get the value of \[0.\overline {23} \] in \[\dfrac{p}{q}\] form.

Complete step-by-step answer:
We will solve this question by forming two equations and subtracting them. The purpose of this is to remove the numbers in the decimal places.
The bar sign above 23 indicates that the numbers 23 repeats in the decimal expansion.
Let \[x = 0.23232323\].
Here, we can observe that 23 repeats in the decimal expansion.
Since 23 is two digits, we will multiply the expression by 100 to find the value of \[100x\].
Multiplying both sides of the expression by 100, we get
\[100x = 23.232323\]
We will extend the decimal places such that the number of decimal places in the value of \[x\] and \[100x\] is the same.
The number of decimal places in \[x = 0.23232323\] is 8.
Thus, extending the decimal places in \[100x = 23.232323\] from 6 to 8, we get
\[100x = 23.23232323\]
Next, we need to subtract the two equations such that numbers in the decimal places are 0.
Subtracting the equation \[x = 0.23232323\] from the equation \[100x = 23.23232323\], we get
\[100x - x = 23.23232323 - 0.23232323\]
Subtracting the terms in the equation, we get
\[\begin{array}{l} \Rightarrow 99x = 23.00000000\\ \Rightarrow 99x = 23\end{array}\]
The decimal places are removed from the expression.
Now, we will divide both sides by 99 to get the value of \[x\] in the \[\dfrac{p}{q}\] form.
Dividing both sides by 99, we get
\[\begin{array}{l} \Rightarrow \dfrac{{99x}}{{99}} = \dfrac{{23}}{{99}}\\ \Rightarrow x = \dfrac{{23}}{{99}}\end{array}\]
\[\therefore\] The number \[0.\overline {23} \] is converted into the \[\dfrac{p}{q}\] form, \[\dfrac{{23}}{{99}}\].

Note: Here, we can see that there is a bar above \[0.23\], this means that 23 is a recurring number. We are asked to find \[\dfrac{p}{q}\] form of \[0.\overline {23} \] where \[p\] and \[q\] are integers and \[q \ne 0\], this means we need to find the rational number. Rational number is a fraction whose denominator is not equal to zero.