Question

How do you convert $ - 6.3$$(3$repeating$)$ to a fraction?

Answer 1

Hint: In this question, we are given a number and we have been asked to convert the number into fraction. First, we will consider $x$ as the given number.
$x = - 6.\overline 3 $
Now, we will multiply the given number by $10$ and $100$ . Then, we will subtract both the values. After that, shift the coefficient of the variable to the other side and find the simplified value. Hence, you will get your answer.

Complete step by step answer:
$ - 6.3$ with $3$ repeating can be written as \[ - 6.\overline 3 \] . Where, the bar on top of the $3$ means a repeating number or pattern of numbers.
Now, consider
$x = - 6.\overline 3 $
Now, we will multiply both the sides by $10$ .
$10x = - 63.\overline 3 $ …. (1)
Now, we will multiply both the sides by $100$ .
$100x = - 633.\overline 3 $ …. (2)
Now, we will subtract equation (1) from equation (2)
Then, we will get,
$100x - 10x = - 633.\overline 3 - \left( { - 63.\overline 3 } \right)$
Simplifying further, we get,
$90x = - 633.\overline 3 + 63.\overline 3 $
Simplifying further,
$90x = - 570$
Now, we will divide both the sides by $90$ .
$\dfrac{{90}}{{90}}x = - \dfrac{{570}}{{90}}$
Now, we will cancel the like terms on left-hand side and simplify on the right-hand side,
$x = - \dfrac{{57}}{9} = - \dfrac{{19}}{3}$
Hence, from our operations, we have got $x = - 6.\overline 3 = - \dfrac{{19}}{3}$ .

Note: Alternative method for this problem is,
First, we can write what is given to us –
$x = - 6.\overline 3 $ …. (1)
Next, we will multiply each side by $10$ , we will get,
$10x = - 63.\overline 3 $ …. (2)
Now, we will subtract equation (1) from equation (2)
Then, we will get,
$10x - x = - 63.\overline 3 - ( - 6.\overline 3 )$
Now add the second term, hence we get
$10x - x = - 63.\overline 3 + 6.\overline 3 $
On subtracting both the sides, we will get,
$9x = - 57$
Now, we will divide both the sides by $9$ .
$\dfrac{9}{9}x = - \dfrac{{57}}{9}$
Now, we will cancel the like terms on left-hand side and simplify on the right-hand side,
$x = \dfrac{{19}}{3}$
Hence, from our operations, we have got $x = - 6.\overline 3 = - \dfrac{{19}}{3}$ .