Question

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

Answer 1

Hint: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.