Question

How do you convert \[ - 6.\overline 3 \](3 repeating) to a fraction?

Answer 1

Hint: In this question we have to convert \[ - 6.\overline 3 \] into a fraction, to do this we have to consider the given decimal as a variable i.e., \[ - 6.\overline 3 = x\], then we have to multiply and divide \[ - 6.\overline 3 \] by 10, by doing so we will get an equation, then again multiply and divide \[ - 6.\overline 3 \] 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 decimal is \[ - 6.\overline 3 \], 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.,
\[x = - 6.\overline 3 \],
This can be rewritten as,
\[x = - 6.3333.....\]
Now multiply and dividing by 10, we get,
\[x = - 6.3333..... \times \dfrac{{10}}{{10}}\],
Now simplifying we get,
\[10x = - 6.3333..... \times 10\],
Now multiplying we get,
\[10x = - 63.333..... - - - - - (1)\],
Now multiply and divide the given decimal by 100, we get,
\[x = - 6.3333..... \times \dfrac{{100}}{{100}}\],
Now simplifying we get,
\[100x = - 6.3333..... \times 100\],
Now multiplying we get,
\[10.x = - 633.3333..... - - - - - (2)\],
Now solving the two equations (1) and (2) by subtracting (1) from (2), we get,
\[100x - 10x = - 633.3333..... - ( - 63.3333)\],
Now simplifying we get,
\[ \Rightarrow 90x = - 633.3333 + 63.3333\],
So, now by subtracting we get,
\[ \Rightarrow 90x = - 570\],
Now dividing both sides with 90, we get,
\[ \Rightarrow \dfrac{{90x}}{{90}} = \dfrac{{ - 570}}{{90}}\],
Now simplifying we get,
\[ \Rightarrow x = \dfrac{{ - 57}}{9}\],
So we know that \[x = - 6.\overline 3 \], so the fraction of the given decimal is \[\dfrac{{ - 57}}{9}\].

\[\therefore \]The converted fraction for the given decimal \[ - 6.\overline 3 \] is equal to \[\dfrac{{ - 57}}{9}\].

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.