Question

How do you convert $0.\bar 4\bar 9$ ($49$ repeating ) as a fraction?

Answer 1

Hint: In order to convert $0.\bar 4\bar 9$ i.e., a repeating decimal into a fraction, we will consider $x = 0.494949 \ldots $. Then, multiply it with $10$ and $1000$ respectively. So, we will get two equations, where we will subtract the equation $\left( 1 \right)$ from $\left( 2 \right)$. And, by evaluating it we will determine the required fraction.

Complete step by step solution:
Now, we want to convert $0.\bar 4\bar 9$ into a fraction.
We know that the repeating decimals are called rational numbers. Thus, it can be converted into the fraction form.
Let $x = 0.494949 \ldots $
Let us multiply and divide by $10$, we have,
$x = 0.494949 \ldots \times \dfrac{{10}}{{10}}$
$x = \dfrac{{4.9494 \ldots }}{{10}}$
$10x = 4.9494 \ldots $ $ \to \left( 1 \right)$
Now, let us multiply and divide by $1000$, we have,
$x = 0.494949 \ldots \times \dfrac{{1000}}{{1000}}$
$x = \dfrac{{494.9494 \ldots }}{{1000}}$
$1000x = 494.9494 \ldots $ $ \to \left( 2 \right)$
Now, subtract equation $\left( 1 \right)$ from $\left( 2 \right)$, we have,
$1000x - 10x = 494.9494 \ldots - 4.9494 \ldots $
$990x = 490$
$x = \dfrac{{490}}{{990}} = \dfrac{{49}}{{99}}$

Hence, the value of $0.\bar 4\bar 9$in terms of fraction is $\dfrac{{49}}{{99}}$.

Note: Repeating decimals are those numbers which keep on repeating the same value after decimal point. These numbers are also known as Recurring numbers. The common definition of rational number that is known is that any number that can be written in fraction form is a rational number.

Here we have multiplied and divided $0.\bar 4\bar 9$ by $10$ and $1000$ respectively, then subtracted both the equations to determine the value of $x$ as in this question we have a repetition of $49$ in $0.\bar 4\bar 9$. Normally, to convert a decimal to a fraction, place the decimal number over its place value. For example, if we have $0.4$, the $4$ is in the tenth place, so we place $4$ over $10$ to create the equivalent fraction, i.e., by multiply and dividing by $10$, we have $\dfrac{4}{{10}}$. If we have two numbers after the decimal point, then we use $100$, if there are three then we use $1000$, etc.