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# What is the value of $0.\overline {57}$A. $\dfrac{{57}}{{10}}$B. $\dfrac{{77}}{{99}}$C. $\dfrac{{19}}{{33}}$D. $\dfrac{{52}}{9}$

Last updated date: 19th Mar 2023
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Hint:In order to find the given value, we should first know what this bar is over $57$. This bar is known as the bar notation, which says that the numbers having a bar over it is being repeated over and over again. For example: $0.\overline x = 0.xxxxxxxxxxx........$ and it continues.

Complete step by step answer:
Naming the given value as $x$; so, it becomes $x = 0.\overline {57}$.
Since, it is a bar notation, so the value can be written as
$x = 0.57575757575757.....$………..(1)
Since the bar is on two consecutive digits and we need the same value after decimal, we are moving the decimal point towards right by two digits.
For that we need to multiply the value (1) by $100$.
So, multiplying the equation (1) by $100$, and we get:
$100x = 100 \times \left( {0.57575757575757.....} \right)$
$\Rightarrow 100x = 57.575757575757......$………(2)

Since, we need to simplify the terms in order to get the value of $x$. So, subtracting (1) from (2), and we get:
$100x - x = \left( {57.575757575757......} \right) - \left( {0.575757575757......} \right)$
Solving the left-hand side and the right side separately, and we can see that the values after decimals are the same and can be cancelled out from simple subtraction. So, we get:
$100x - x = \left( {57.575757575757......} \right) - \left( {0.575757575757......} \right) \\ \Rightarrow 99x = 57 \\$
Dividing both the sides by $99$ in order to get the value of $x$ , and we get:
$\Rightarrow \dfrac{{99x}}{{99}} = \dfrac{{57}}{{99}} \\ \Rightarrow x = \dfrac{{57}}{{99}} \\$
Since, we can see that the value on the right-side can be further simplified, by dividing both the numerator and denominator by $3$ as it’s the highest factor that can divide both of them. So, dividing and multiplying the right-side by$3$, we get:
$\Rightarrow x = \dfrac{{\dfrac{{57}}{3}}}{{\dfrac{{99}}{3}}} \\ \Rightarrow x = \dfrac{{19}}{{33}} \\ \therefore x = 0.\overline {57}$

Therefore, the value of $0.\overline {57}$ and the correct answer is option C.

Note:Do not commit a mistake by repeating only $5$ or $7$ from $0.\overline {57}$, as because the bar is upon both $5$ and $7$, so must be repeated in the same order. Since, the bar was on two digits that’s why we multiplied the value by $100$, in order to take the two digits out. If the bar is on one digit then we can multiply by $10$, then subtract.