Question

What is $ -0.5$ repeating as a fraction?

Answer 1

Hint: For this question, we write the number -0.5 as a repeating decimal which is given as $-0.\overline{5}=-0.5555\ldots \ldots $ We then consider this number as x, that is $x=-0.555\ldots $ Then we multiply both sides of this equation by 10. Now we subtract both the equations and divide both sides of the equation by 9 to get the value of x which can be used to obtain the solution.

Complete step by step solution:
We are required to find the number -0.5 repeating as a fraction. In order to do so, we need to consider the number as a recurring number or repeating decimal. This can be represented as $-0.\overline{5}=-0.5555\ldots $ Now we equate this to a variable, say x,
$\Rightarrow x=-0.5555\ldots \ldots \left( 1 \right)$
We now multiply both sides of this equation by 10.
$\Rightarrow 10x=10\times -0.5555$
Multiplying the terms on the right-hand side, we can see that the decimal point shifts right by 1 place.
$\Rightarrow 10x=-5.5555\ldots \ldots \left( 2 \right)$
We need to find the solution for x from both these equations. This can be done by subtracting the two equations.
$\Rightarrow \left( 2 \right)-\left( 1 \right)$
$\Rightarrow 10x-x=-5.5555-\left( -0.5555 \right)$
Subtracting x from 10x on the left-hand side and simplifying the two numbers on the right-hand side,
$\Rightarrow 9x=-5.5555+0.5555$
Adding 0.5555 to -5.5555,
$\Rightarrow 9x=-5$
Dividing both sides of the equation by 9,
$\Rightarrow x=\dfrac{5}{9}$
Therefore, since x is nothing but the repeating number, we have represented the given number in fraction as shown.
Hence, -0.5 repeating has been represented as a fraction $\dfrac{5}{9}.$

Note: To solve and understand this type of questions, students need to know what rational numbers are. The concepts of recurring or non-terminating numbers are used in this question. The main objective to solve this question is by making the numbers after the decimal point 0. In order to do so, we can even multiply both sides of the equation initially by 100 too and subtract.