Question

Add $3.8\overline{92}$ and $2.0\bar{7}$ and express the sum as a rational number.

Answer 1

Hint: We are given two non-terminating repeating decimals. The bar over a number indicates that the number is repeating infinite times. We will first express the given decimal form in fractional form. Then, we will add those numbers to get the required answer.

Complete step-by-step answer:
We will begin by converting the decimal in fractional form.
Let the number $3.8\overline{92}$ be $x$
There is one number after the decimal in the expression, $3.8\overline{92}$ which has no bar over it.
Hence, we will multiply 10 on both sides,
$10x=38.\overline{92}$ eqn. (1)
Now, there are two numbers that have bar over them
We will multiply equation (1) by 100.
$1000x=3892.\overline{92}$ eqn. (2)
Subtract equation (2) from (1)
$$\begin{gathered} 1000x-10x=3892.\overline{92}-38.\overline{92} \\ \Rightarrow 990x=3854 \end{gathered}$$
Divide both sides by 990
$x={\displaystyle \frac{3854}{990}}$
Therefore, the number $3.8\overline{92}$ is equal to ${\displaystyle \frac{3854}{990}}$
Let the number $2.0\bar{7}$ be $y$.
There is one number after the decimal in the expression, $2.0\bar{7}$ which has no bar over it.
Hence, we will multiply 10 on both sides,
$10y=20.\bar{7}$ eqn. (3)
Now, there are one numbers that have bar over them
We will multiply equation (1) by 10.
$100y=207.\bar{7}$ eqn. (4)
Subtract equation (4) from (3)
$$\begin{gathered} 100y-10y=207.\bar{7}-20.\bar{7} \\ \Rightarrow 90y=187 \end{gathered}$$
Divide both sides by 90
$y={\displaystyle \frac{187}{90}}$
Therefore, the number $2.0\bar{7}$ is equal to ${\displaystyle \frac{187}{90}}$
We have to add both the numbers.
$$\begin{gathered} {\displaystyle \frac{3854}{990}}+{\displaystyle \frac{187}{90}} \\ \Rightarrow {\displaystyle \frac{3854+2057}{990}} \\ \Rightarrow {\displaystyle \frac{5911}{990}} \end{gathered}$$

Note: The number which has decimal of the form which is terminating, non-terminating, repeating can be expressed as rational numbers. Whereas the numbers which have non-terminating, non-repeating decimal expansion are irrational numbers.