Question

How do you convert $1.0\overline{1}$ (here $1$ is being repeated) to a fraction?

Answer 1

Hint: We can easily solve problems on conversions of repeating decimals to fractions by an algebraic method in just few steps. To do that we assume the number that is given to be x. Since x is recurring in $1$ decimal places, we multiply it by $10$ . Upon doing that we subtract $x$ from $10x$ and after dividing the result of the subtraction by $9$ we get the fractional form of the given number.

Complete step by step solution:
The number we are given is
\[1.0\overline{1}\] (here $1$ is being repeated)
We can convert this number into its fractional form by the algebraic method in few steps.
We assume the given number of repeating decimals to be x as
\[x=1.0\overline{1}\]
As the number x has repeating decimals upto one digits, we multiply it with the power $1$ of $10$ i.e., $10$ as shown below
$\Rightarrow 10x=10.\overline{1}$
Now to eliminate the repeating decimals we must subtract x from the above expression i.e., $10x$ as shown below
$\Rightarrow 10x-x=10.\overline{1}-1.0\overline{1}$
$\Rightarrow 9x=10.1-1.0$
$\Rightarrow 9x=9.1$
Further dividing both sides of the above equation by $9$ we get
$\Rightarrow x=\dfrac{9.1}{9}$
Further, multiplying both the numerator and denominator of the right hand side with \[10\] we get
$\Rightarrow x=\dfrac{91}{90}$
Therefore, we conclude that the given number \[1.0\overline{1}\] (here $1$ is being repeated) can be written in fraction form as $\dfrac{91}{90}$ .

Note: We must keep in mind that the entire calculation of this type of problems must be done properly so that inaccuracy can be avoided. Also, we have to see how many digits after the decimal points are being repeated from which we will decide the power of $10$ to be multiplied with x, which is in this case $1$ .