Question

How do we convert \[0.\overline {15} \](15 being repeating) to a fraction?

Answer 1

Hint: In this question we have to convert \[0.\overline {15} \] into a fraction, to do so we have to consider the given decimal as a variable i.e., \[0.\overline {15} = x\], then we have to multiply and divide \[0.\overline {15} \] by 1, by doing so we will get an equation, then again multiply and divide \[0.\overline {15} \] by 100, by doing so we will get another equation, then by doing subtraction we will get the required converted fraction.

Complete step-by-step solution:
Given a decimal is \[0.\overline {15} \], we have to convert the decimal into a fraction.
We know that the repeating decimals are called rational numbers. Thus it can be converted into the fraction.
Let us consider the given decimal as a variable, i.e.,
\[x = 0.\overline {15} \],
This can be rewritten as,
\[ \Rightarrow \]\[x = 0.15151515.........\]
Now multiply and dividing by 10, we get,
\[ \Rightarrow x = 0.15151515.... \times 1\],
Now simplifying we get,
\[ \Rightarrow x = 0.151515..... - - - - - (1)\],
Now multiply and divide the given decimal by 100, we get,
\[ \Rightarrow x = 0.151515..... \times \dfrac{{100}}{{100}}\],
Now simplifying we get,
\[ \Rightarrow 100x = 0.151515..... \times 100\],
Now multiplying we get,
\[ \Rightarrow 100x = 15.1515...... - - - - - (2)\],
Now solving the two equations (1) and (2) by subtracting (1) from (2), we get,
\[ \Rightarrow 100x - 1x = 15.151515..... - 0.151515.....\],
Now simplifying we get,
\[ \Rightarrow 99x = 15.151515.... - 0.151515.....\],
Now simplifying by subtracting we get,
\[ \Rightarrow 99x = 15\],
Now dividing both sides with 99, we get,
\[ \Rightarrow \dfrac{{99x}}{{99}} = \dfrac{{15}}{{99}}\],
Now simplifying we get,
\[ \Rightarrow x = \dfrac{{15}}{{99}}\],
Now simplifying the fraction we get,
\[ \Rightarrow x = \dfrac{5}{{33}}\],
So we know that \[x = 0.\overline {15} \], so the fraction of the given decimal is \[\dfrac{5}{{33}}\].

\[\therefore \]The converted fraction for the given decimal \[0.\overline {15} \] is equal to \[\dfrac{5}{{33}}\].

Note: Repeating decimals are the numbers which will have the repeating value after the decimal point. These numbers are called Recurring numbers. The definition of a rational number that is known is that any number that can be written in fraction form is called a rational number.