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Express $0.4\overline{57}$ in the form of $\dfrac{p}{q},\text{ where }q\ne 0$ ?

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Last updated date: 20th Jun 2024
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Answer
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Hint: In this question, we have to express the recurring decimal in the form of fractional numbers. As we know, the recurring decimal is a type of decimal in which the decimal numbers are repeated. So, we will represent the recurring numbers by putting the bar signs on the repeated number. Also, fractional numbers are those numbers which are expressed as the form of $\dfrac{p}{q}$ , where p and q are integers and $q\ne 0$ . So, we will first let the given number equal to x and then find the number of recurring digits which is equal to n. Then, we will multiply ${{10}^{n}}$ in the equation. After that, we will subtract the equations $x=0.45757...$ and $100x=45.75757...$ . In the end, we will divide the equation by 99 and make the necessary calculations, to get the required solution for the problem.

Complete step by step solution:
According to the question, we have to find the fractional number from a recurring decimal.
The recurring decimal given to us is $0.4\overline{57}$
Now, we will first let the given recurring decimal equal to x, we get
$x=0.45757...$ --------- (1)
Now, we will calculate the number of recurring decimal which is equal to n in the given problem, we get that 57 are the number that has the bar on them, implies these two digits are repeated in the given recurring decimal, therefore we get
$n=2$
Now, we will multiply ${{10}^{n}}$ on both sides in the equation (1), we get
$x\times {{\left( 10 \right)}^{n}}=0.45757...\times {{\left( 10 \right)}^{n}}$
As $n=2$ , thus we get
$x\times {{\left( 10 \right)}^{2}}=0.45757...\times {{\left( 10 \right)}^{2}}$
On further simplification, we get
$100x=45.75757...$ -------- (2)
Now, we will subtract equation (1) and (2), we get
$100x-x=45.75757...-0.45757...$
Therefore, we get
$99x=45.3$
Now, we will divide 99 on both sides in the above equation, we get
$\dfrac{99x}{99}=\dfrac{45.3}{99}$
Now, we will remove the decimal sign by placing 1 zero in the denominator, we get
$x=\dfrac{453}{990}$

Note: While solving this problem, do the step-by step calculations to avoid confusion and mathematical error. Do not forget what is $n$ in the problem, it is the number of the recurring digits in the recurring decimal number, thus calculate it properly to get an accurate solution.