Question

Show that $0.3333.... = 0.\overline 3 $ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$.

Answer 1

Hint: To write a recurring decimal as a fraction in simplest form, we need to suppose that decimal as $x$ and then multiply it by $10$ and then subtract $x$ from the result. On solving the obtained equation, we will get the fraction form of the given recurring decimal.

Complete step by step solution:
In this question, we are given a recurring decimal and we need to express in fraction form that is $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q$ is not equal to $0$.
Given a recurring decimal is: $0.3333.... = 0.\overline 3 $
We are given a recurring decimal and we need to express the decimal as a fraction in simplest form.
Now, we know that a recurring decimal is a rational number and it can be converted to fraction form.
First of all, let us suppose this recurring decimal as $x$.
$ \Rightarrow x = 0.3333.... - - - - - - \left( 1 \right)$
Now, to eliminate the digits after the decimal point, we need to multiply $x$ with $10$.
Multiplying equation $\left( 1 \right)$ with $10$, we get
$10x = 3.3333.... - - - - - \left( 2 \right)$
Now, we have to subtract $x$ from the result.
So, equation $\left( 2 \right) - $$\left( 1 \right)$gives
$
  10x - x = 3.3333... - 0.3333... \\
\Rightarrow 9x = 3 \\
\Rightarrow x = \dfrac{3}{9} = \dfrac{1}{3} \\
 $

Hence, we have written $0.3333...$ as a fraction in the form $\dfrac{p}{q}$.

Note:
We can solve this question using an alternate method also. In this method, we can use a direct formula. The formula is
$ \Rightarrow \dfrac{{\left( {Decimal \times F} \right) - \left( {Non - repeating\,part} \right)}}{D}$
Here,
$F = 10$, if only one digit is repeating.
$F = 100$, if two digits are repeating.
$D = 9$, if one digit is repeating.
$D = 99$, if two digits are repeating.
In our question, only one digit that is 3 is repeating.
So, $F = 10,D = 9$ and the non-repeating part is 0.
Putting this values in the formula we get,
Fraction form of $0.3333...$$ = \dfrac{{\left( {0.3 \times 10} \right) - 0}}{9}$
$
   = \dfrac{{3 - 0}}{9} \\
   = \dfrac{3}{9} \\
   = \dfrac{1}{3} \\
 $