Question

How do you write the repeating decimal \[0.\bar 5\] as a fraction?

Answer 1

Hint: In order to convert \[0.\bar 5\] i.e., a repeating decimal into a fraction, we will consider $ x = 0.555 \ldots $ . Then, multiply it with $ 10 $ and $ 100 $ respectively. So, we will get two equations, where we will subtract the equation $ \left( 1 \right) $ from $ \left( 2 \right) $ . And, by evaluating it we will determine the required fraction.

Complete step-by-step answer:
Now, we want to convert \[0.\bar 5\] into a fraction.
We know that the repeating decimals are called rational numbers. Thus, it can be converted into the fraction form.
Let $ x = 0.555 \ldots $
Let us multiply and divide by $ 10 $ , we have,
 $ x = 0.555 \ldots \times \dfrac{{10}}{{10}} $
 $ x = \dfrac{{5.555 \ldots }}{{10}} $
 $ 10x = 5.555 \ldots $ $ \to \left( 1 \right) $
Now, let us multiply and divide by $ 100 $ , we have,
 $ x = 0.555 \ldots \times \dfrac{{100}}{{100}} $
 $ x = \dfrac{{55.555 \ldots }}{{100}} $
 $ 100x = 55.555 \ldots $ $ \to \left( 2 \right) $
Now, subtract equation $ \left( 1 \right) $ from $ \left( 2 \right) $ , we have,
 $ 100x - 10x = 55.555 \ldots - 5.555 \ldots $
 $ 90x = 50 $
 $ x = \dfrac{{50}}{{90}} $
 $ x = \dfrac{5}{9} $
Hence, the value of \[0.\bar 5\]in terms of fraction is $ \dfrac{5}{9} $ .
So, the correct answer is “$ \dfrac{5}{9} $”.

Note: Here we have multiplied and divided \[0.\bar 5\] by $ 10 $ and $ 100 $ respectively, then subtracted both the equations to determine the value of $ x $ as in this question we have a repetition of $ 5 $ in \[0.\bar 5\]. Normally, to convert a decimal to a fraction, place the decimal number over its place value. For example, if we have $ 0.5 $ , the $ 5 $ is in the tenth place, so we place $ 5 $ over $ 10 $ to create the equivalent fraction, i.e., by multiply and dividing by $ 10 $ , we have \[\dfrac{5}{{10}}\]. If we have two numbers after the decimal point, then we use $ 100 $ , if there are three then we use $ 1000 $ , etc.