Question

Show that $0.2353535... = 0.2\overline {35} $ can be expressed in the form of $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$ .

Answer 1

Hint:In this question we have to prove that the above given number
$0.2353535... = 0.2\overline {35} $ can be expressed in the form of $\dfrac{p}{q}$. So we will first assume that let $x$ be equal to $0.2353535...$, i.e. $x = 0.2353535... = 0.2\overline {35} $. This will be our first equation. After that we will multiply both sides of the equation by $100$ and then we will subtract the second equation from the first equation to get the required value.

Complete step by step answer:
Here we have been given the definition of a rational number. We know that a rational number is a number that can be expressed as the fraction of
$\dfrac{p}{q}$ , where $p$ and $q$ , both are integers and $q \ne 0$ .
An example of a rational number is
$\dfrac{{ - 3}}{7}$ .
Now let us assume that $x = 0.2353535... = 0.2\overline {35} $ .
This is the first equation.
We will multiply both the sides by $100$ :
$100 \times x = 100 \times 0.2353535...$
Upon multiplication, it gives us the value:
$100x = 23.53535...$
It is the second equation.
Now we will subtract the first equation from the second equation, i.e.
$100x - x = 23.53535... - 0.2353535$
It gives us the value;
$99x = 23.2999...$
We can round off the value in the right hand side of the equation, so it can be written as
$99x = 23.3$
By isolating the term $x$ in the left hand side, it gives:
$\therefore x = \dfrac{{23.3}}{{99}}$
Hence we get the required value in the form of $\dfrac{p}{q}$, where
$p = 23.3,q = 99$ and $q \ne 0$ .

Note:We should note that there are two types of rational numbers: Terminating numbers and Non terminating but repeating numbers. We should know that a decimal fraction or a number that will never come to an end but will repeat one or more numbers after the decimal point are called non terminating but repeating rational numbers.So in the above question, the number $0.2353535...$ is a non-terminating but repeating decimal. It is denoted by the bar on two numbers that are repeating i.e. $0.2\overline {35} $.