Question

How do you find the repeating decimal $0.1$ with $1$ repeated as a fraction?

Answer 1

Hint: Start with assuming the given number as $m = 0.1111..$. This remains our original equation. Now you can obtain a new equation by multiplying both sides by $10$. You can now subtract the original equation from the new equation to obtain a value for ‘m’ that is not in the form of decimals.

Complete step by step answer:
Here in this question, we are given a decimal number $0.1111111.....$ which is a non-ending repeating decimal number. And we need to change this number in its fraction form.
Before starting with the solution to this problem, we must understand a few concepts related to this question. A rational number is a number that can be expressed in the form of $\dfrac{p}{q}$ where $p$ and $q$ are both integers and the $q$ can never be equal to $0$. These rational numbers can be changed in either their terminating or non-terminating decimal form. Now, non-terminating decimal numbers can be further classified into two types which are recurring and non-recurring decimal numbers. Recurring numbers are those numbers that keep on repeating the same value after the decimal point. These numbers are also known as repeating decimals.
Also, every decimal number can be changed back to its rational form, except non-repeating recurring numbers because they are irrational numbers and can’t be expressed as $\dfrac{p}{q}$.
So, let’s assume the given number as:
$ \Rightarrow m = 0.111111....$
Now multiply both sides of the equation with $10$, which will give us:
$ \Rightarrow m \times 10 = 10 \times 0.111111.... \Rightarrow 10m = 1.11111....$
This gives us two equations, one with the original number and the other with a different number but with the same decimal part. Let’s subtract the original equation from the newly obtained equation
$ \Rightarrow 10m - m = \left( {1.11111....} \right) - \left( {0.11111....} \right)$
On solving the RHS and LHS separately, we get:
$ \Rightarrow 10m - m = \left( {1.11111....} \right) - \left( {0.11111....} \right) \Rightarrow 9m = 1$.
Therefore, we got an equation in the variable $'m'$ without the repeating decimal digits. Now we can transform the equation to get the value of unknown:
$ \Rightarrow 9m = 1 \Rightarrow m = \dfrac{1}{9}$
Thus we get the value of unknown as: $m = \dfrac{1}{9}$
But we already assume the value of $m$ in the original equation, i.e. $m = 0.111111....$
$ \Rightarrow m = 0.1111.... = \dfrac{1}{9}$

Thus the required fraction form of the number is $0.11111....$ is $\dfrac{1}{9}$.

Note: In this question, the assumption $m = 0.111...$ played a crucial role in the solution of the problem. The method here used $10$ to multiply in the equation $m = 0.111111....$ but you can use any multiple of $10$ in this case to do the same task. That is, $100m = 11.111111....$ can be obtained by multiplying $100$, and then the original equation can be subtracted from it to get the required answer.