Question

Convert \[0.9\] (\[9\] being repeated) into a fraction.

Answer 1

Hint: A fraction represents a part of a whole. To convert the given expression to a fraction, first multiply it by \[10\], then subtract the original number from the new number obtained, then make necessary conversion and substitutions to obtain the result.

Complete step by step solution:
In the given decimal the number \[9\] keeps on repeating, that is it is a recurring decimal.
Observe that \[0.9999....\] can be represented as \[0.\bar 9\]
First multiply \[0.\bar 9\] with \[10\]:
\[0.\bar 9 \times 10 = 9.\bar 9\] ……… (1)
Now, subtract \[0.\bar 9\] from \[9.\bar 9\]
\[\therefore \] \[9.\bar 9 - 0.\bar 9\] \[ = \] \[9\]..(2)
Substitute the value of \[9.\bar 9\] from equation (1) into equation (2):
\[\left( {0.\bar 9 \times 10} \right)\] \[ - \] \[0.\bar 9\] \[ = \] \[9\]
Take \[0.\bar 9\] common in left hand side of the equation:
\[ \Rightarrow \] \[0.\bar 9\left( {10 - 1} \right) = 9\]
\[ \Rightarrow \] \[0.\bar 9 \times 9 = 9\]
Dividing both sides of the equation by \[9\]:
\[ \Rightarrow \] \[0.\bar 9 = \dfrac{9}{9}\]
Remove common factor of \[9\]:
\[ \Rightarrow \]\[0.\bar 9 = \dfrac{1}{1} = 1\]
Hence \[0.9999.... = \dfrac{1}{1}\].

Additional Information: 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: \[0.\dot 7\] , \[0.\mathop {82}\limits^{\_\_} \]

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 if we have to express \[0.\bar 6\] as a fraction, then 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 obtained by following the above described method that is \[\dfrac{6}{9}\], the latter is much more accurate.