Question

How do you convert $-1.\overline{3}$ (3 repeatings) to a fraction?

Answer 1

Hint: In the above question we were asked to convert -1.3 to a fraction. We know that the total digits after the decimal point is equal to the number of zeroes in the denominator. We also need to take care of the minus sign. So let us see how we can solve this problem.

Complete Step by Step Solution:
We have to convert $-1.\overline{3}$ (3 repeatings) into a fraction.
Let, x = $- 1.\mathop 3\limits^.$, the point on the above 3 indicates that the digit is repeating
As it is repeating 3 so,
First, we find the value of 10x
 $\Rightarrow 10x = - 13.3$
Then we will find the value of 9x
 $\Rightarrow 10x - x = - 13.3 - ( - 1.3)$
Minus-Minus is plus
 $\Rightarrow 9x = - 13.3 + 1.3$
Therefore, 9x is
 $\Rightarrow 9x = - 12$
After dividing both sides with 9 we get,
 $\Rightarrow x = - \dfrac{{12}}{9}$
 After solving this we get,
 $\Rightarrow x = - \dfrac{4}{3}$

Therefore, -1.3 (3 repeating) is $- \dfrac{4}{3}$ when converted into a fraction.

Note:
In the above problem, it is mentioned that 1.3 (3 repeatings). For solving these types of repeating decimals, we suppose the repeating portion is n digit long for this problem it is 3. Then we let the initial value as x and we find x from ${10^n}x - x$.