Question

How do you convert $- 0.1\bar 3$ ( $3$ being repeated) to a fraction?

Answer 1

In this question, we need to convert $- 0.1\bar 3$ into fraction. Here, we will consider $- 0.1\bar 3$ as $a$ . Then we will multiply and divide $- 0.1\bar 3$ by $10$ . By which we will get an equation, mark it as equation (1). Then we will multiply and divide $- 0.1\bar 3$ by $100$ . By which we will get another equation, mark it as equation (2). And, subtract equation (1) from equation (2), then by evaluating it we will get the required fraction.

Complete step-by-step solution:
In this question, we need to convert $- 0.1\bar 3$ to a fraction.
Let $a$ be the fraction that we required.
Here, let us consider $- 0.1\bar 3$ .

Now, multiply and divide $- 0.1\bar 3$ by $10$ , we have,
$a = - 0.13333... \times \dfrac{{10}}{{10}}$
Then, $a = \dfrac{{ - 0.1333...}}{{10}}$
Hence, $10a = - 1.333...$

Let us consider this as equation (1).

Now, let us multiply and divide $- 0.1\bar 3$ by $100$ , we have,
$a = - 0.1333... \times \dfrac{{100}}{{100}}$
Then, $a = \dfrac{{ - 13.333...}}{{100}}$

Hence, $100a = 13.333...$

Here, let us consider this as equation (2).

Now, let us subtract equation (1) from equation (2).

Therefore, we have,
$100a - 10a = - 13.333... - \left( { - 1.333...} \right)$
$90a = - 13.333... + 1.333...$
Hence, $90a = - 12$

So, $a = \dfrac{{ - 12}}{{90}}$
Therefore, $a = \dfrac{{ - 2}}{{15}}$

Hence, the converted value of $- 0.1\bar 3$ to a fraction is $\dfrac{{ - 2}}{{15}}$ .

Note: In this question it is important to note that, here we have multiplied and divided $- 0.1\bar 3$ by $10$ and $100$ respectively, then subtracted both the equations to determine the value of $a$ as in this question we have a repetition of $3$ in $- 0.1\bar 3$ . Normally, to convert a decimal to a fraction, place the decimal number over its place value. For example, if we have $0.1$ , the $1$ is in the tenth place, so we place $1$ over $10$ to create the equivalent fraction, i.e., by multiplying and dividing by $10$ , we have $\dfrac{1}{{10}}$ . If we have two numbers after the decimal point, then we use $100$ , if there are three then we use $1000$ , etc.