Question

What is $0.5$ with the $5$ recurring as a fraction?

Answer 1

Hint: $0.5$ recurring means $0.5555....$ .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:
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 = 0.5555.... - - - - - - \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
$10x = 5.5555.... - - - - - \left( 2 \right)$
Now, we have to subtract $x$ from the result.
So, equation $\left( 2 \right) - $$\left( 1 \right)$gives
$
  10x - x = 5.5555... - 0.5555... \\
  9x = 5 \\
  x = \dfrac{5}{9} \\
 $
This is our final answer.
Hence, we have written $0.5$ recurring as $\dfrac{5}{9}$ a fraction in simplest form.

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 $5$ is repeating.
So, $F = 10,D = 9$ and the non-repeating part is 3.
Putting this values in the formula we get,
Fraction form of $0.555...$$ = \dfrac{{\left( {0.5 \times 10} \right) - 0}}{9}$
 $
   = \dfrac{{5 - 0}}{9} \\
   = \dfrac{5}{9} \\
 $