Question

# Express the decimal $0.\overline {621}$ in the form $\dfrac{p}{q}$.

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Hint: $0.\overline {621}$ is a recurring decimal which can be written as $0.621621621......$. Assume it some variable and simplify it to convert in the form of $\dfrac{p}{q}$.

The given decimal is $0.\overline {621}$. Let’s its value is $x$.
Since it’s a recurring decimal, it can be written as $0.621621621......$. So, we have:
$\Rightarrow x = 0.621621621......$
$\Rightarrow 1000x = 621.621621621...., \\ \Rightarrow 1000x = 621 + 0.621621....., \\ \Rightarrow 1000x = 621 + x. \\ \Rightarrow 999x = 621, \\ \Rightarrow x = \dfrac{{621}}{{999}}, \\ \Rightarrow x = \dfrac{{23}}{{37}} \\$
Thus, the $\dfrac{p}{q}$ of $0.\overline {621}$ is $\dfrac{{23}}{{37}}$.
Note: Since the above number is converted in $\dfrac{p}{q}$form, it is called rational number. If the decimal is non-recurring, non-terminating (i.e. continuing endlessly without repetition of any group of digits), then it cannot be converted in the form of $\dfrac{p}{q}$. That’s why such a number is called an irrational number.