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Express each of the following decimals in the form $\frac{p}{q}$:
\( 0 . \overline{621} \)
Given:
The given decimal number is \( 0 . \overline{621} \).
To do:
We have to express the given decimal in $\frac{p}{q}$ form.
Solution:
$0. \overline{621}$
Let $x = 0.621621....$
Multiply both sides by 10.
$10x = 10(0.621621....)$
$10x = 6.21621......$
Multiply both sides by 100.
$100x = 100(0.621621....)$
$100x = 62.1621621.....$
Multiply both sides by 1000.
$1000x = 1000(0.621621....)$
$1000x = 621.621621.....$
Therefore,
$1000x-x = 621.621621.... - 0.621621.....$
$999x = 621$
$x = \frac{621}{999}$
$x=\frac{27\times23}{27\times37}$
$x=\frac{27}{37}$
Therefore,
$0. \overline{621}$ in $\frac{p}{q}$ form is $\frac{27}{37}$.
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