Question

What is the value of $0.\overline{57}$
A). $\dfrac{57}{10}$
B). $\dfrac{57}{99}$
C). $\dfrac{19}{33}$
D). $\dfrac{52}{9}$

Answer 1

Hint: In this question we have to find fraction form of $0.\overline{57}$ for this we will learn concepts of rational numbers and irrational numbers and different types of decimal expansions, whether the given expansion is terminating or non-terminating or recurring or nonrecurring.

Complete step-by-step solution:
Rational number: Rational numbers are the numbers which can be represented in the form of $\dfrac{p}{q}$ where $p\And q$ are integers and $q\ne 0$
Example: $\dfrac{2}{7},\dfrac{11}{15},\dfrac{5}{1},\dfrac{-1}{55}$ are rational numbers but $\dfrac{5}{0}$ is not a rational number
Irrational Number: Irrational numbers are the numbers which cannot be represented in the form $\dfrac{p}{q}$ .
Example: $0.101011011101111011111...$
Every number has a decimal expansion, whether the given number is rational or irrational we can check it by its decimal expansion.
Decimal expansion of a number can be of two types
a). Terminating decimal expansion
b). Non-terminating decimal expansion.
Terminating Decimal Expansion: If in a decimal expansion of a number we will get the remainder $0$ after some decimal digits or we can say that if decimal expansion has a finite number of digits. Then this decimal expansion is called terminating decimal expansion.
Example:
$0.25,11.554,5.85$
$\begin{align}
  & \dfrac{1}{4}=0.25 \\
 & \dfrac{12}{5}=2.4 \\
\end{align}$
These are terminating decimal expansions.
Non terminating decimal expansions: If In a decimal expansion we will never get $0$ as remainder, or we can say that decimal expansion has an infinite number of digits then this type of decimal expansion is called non terminating decimal expansion.
Non terminating decimal expansion are of two types
a) Recurring or repeating non terminating decimal expansion.
b) Non recurring or non-repeating decimal expansion.
Recurring or repeating non terminating decimal expansion: Decimal Expansion which repeats its digits after decimal point are called Recurring or repeating non terminating decimal expansion.
Example:
 :$\begin{align}
  & 0.33333333333...... \\
 & 0.6666666666....... \\
 & 0.1212121212..... \\
\end{align}$
We can also write it as $0.\overline{3},0.\overline{6},0.\overline{12}$
Non recurring or non-repeating decimal expansion: Decimal expansion which does not repeat its digits after decimal expansion are called Non recurring or non-repeating decimal expansion.
Example:
$\begin{align}
  & 0.10100100010000100000.... \\
 & 0.50500500050000500000... \\
\end{align}$
These are Non recurring or non-repeating decimal expansion.
Now let us proceed to our question
We have to find fraction form of $0.\overline{57}$
Let $x=0.\overline{57}$
$\Rightarrow x=0.575757......$ $......eq\left( 1 \right)$
Multiply $eq\left( 1 \right)$ by $10$
$\Rightarrow 10x=5.7575757....$ $........eq\left( 2 \right)$
Multiply $eq\left( 2 \right)$ by $10$
$\Rightarrow 100x=57.575757....$ $........eq\left( 3 \right)$
Subtract $eq\left( 1 \right)\text{ from }eq\left( 3 \right)$
$\Rightarrow 100x-x=57.575757......-0.575757.....$
$\Rightarrow 99x=57$
$\Rightarrow x=\dfrac{57}{99}$
$\therefore 0.\overline{57}=\dfrac{57}{99}$
Hence our answer is option (B).

Note: All terminating decimal expansions are rational numbers and all recurring or repeating decimal expansions are also rational numbers, whereas non terminating non recurring or non-repeating decimal expansions are irrational numbers. There are infinite rational and irrational numbers between any two rational or irrational numbers.