Question

Convert \[0.6\] (\[6\] being repeated) into a fraction.

Answer 1

Hint: A fraction represents a part of a whole. A recurring decimal or repeating decimal is the decimal representation of a number whose digits are repeated periodically and the infinitely repeated portion is not \[0\]. They are represented by putting a bar or a dot above the digit or digits that are periodically repeated.
Examples: \[0.\dot 7\] , \[0.\mathop {82}\limits^{\_\_} \]

Complete step by step solution:
In the given decimal the number \[6\] keeps on repeating itself hence it is a recurring decimal.
\[0.6666....\] can be represented as \[0.\bar 6\]
Multiply \[0.\bar 6\] with \[10\]:
\[0.\bar 6 \times 10 = 6.\bar 6\]..(1)
Subtract \[0.\bar 6\] from \[6.\bar 6\]:
\[6.\bar 6 - 0.\bar 6\] \[ = \] \[6\]..(2)
Substitute the value of \[6.\bar 6\] from equation (1) into equation (2):
\[\left( {0.\bar 6 \times 10} \right)\] \[ - \] \[0.\bar 6\] \[ = \] \[6\]
Take \[0.\bar 6\] common in left hand side of the equation:
\[ \Rightarrow \] \[0.\bar 6\left( {10 - 1} \right) = 6\]
\[ \Rightarrow \] \[0.\bar 6 \times 9 = 6\]
Dividing both sides of the equation by \[9\]:
\[ \Rightarrow \] \[0.\bar 6 = \dfrac{6}{9}\]
\[ \Rightarrow \] \[0.\bar 6 = \dfrac{{2 \times 3}}{{3 \times 3}}\]
Remove common factor of \[3\]:
\[ \Rightarrow \] \[0.\bar 6 = \dfrac{2}{3}\]

Hence \[0.6666....\] \[ = \] \[\dfrac{2}{3}\].

Note: Avoid converting any recurring decimal to a rounded off form and then converting it to a fraction. In that case, the result obtained is less accurate than the result obtained by the described method. For example, in this case, if the decimal is rounded off it becomes \[0.67\] and the fraction becomes \[\dfrac{{67}}{{100}}\], observe that though this fraction is close to the fraction of our answer that is \[\dfrac{6}{9}\], the latter is much more accurate.