Question

How do you write $ 3.5 $ repeating as a fraction in simplest form?

Answer 1

Hint: $ 3.5 $ repeating means $ 3.5555.... $ .To write a repeating 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 repeating decimal.

Complete step-by-step solution:
We are given a repeating decimal and we need to express the decimal as a fraction in simplest form.
Now, we know that a repeating decimal is a rational number and it can be converted to fraction form.
First of all, let us suppose this repeating decimal is $ x $ .
 $ \Rightarrow x = 3.5555.... - - - - - - \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
 $ \Rightarrow 10x = 35.5555.... - - - - - \left( 2 \right) $
Now, we have to subtract $ x $ from the result.
So, equation $ \left( 2 \right) - $ $ \left( 1 \right) $ gives
 $
 \Rightarrow 10x - x = 35.5555... - 3.5555... \\
\Rightarrow 9x = 32 \\
\Rightarrow x = \dfrac{{32}}{9} \\
  $
This is our final answer.

Hence, we have written $ 3.5 $ repeating as $ \dfrac{{32}}{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 $ 3.555... $ $ = \dfrac{{\left( {3.5 \times 10} \right) - 3}}{9} $
$ = \dfrac{{35 - 3}}{9} \\
   = \dfrac{{32}}{9} \\
  $