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Express $2.0 \overline {15}$ in the $\frac{p}{q}$ form, where $p$ and $q$ are integers and $q≠0$.
Given :
The given decimal number is $2.0 \overline {15}$.
To do :
We have to convert $2.0 \overline {15}$ into $\frac{p}{q}$ form.
Solution :
$2.0 \overline {15}$
Let $x = 2.0151515$
Multiplying both sides by 10.
$10x = 10(2.01515....)$
$10x = 20.1515.....$....(i)
$x = 2.0151515$
Multiplying both sides by 100, we get,
$100x = 100(2.01515....)$
$100x = 201.51515.....$....(ii)
$x = 2.0151515$
Multiplying both sides by 1000, we get,
$1000x = 1000(2.01515....)$
$1000x = 2015.1515.....$....(iii)
We can see that, decimal part in $10x$ and $1000x$ is the same, subtracting (i) from (iii), we get,
$1000x-10x=2015.1515......-20.1515.....$
$990x=1995$
$x=\frac{1995}{990}$
$x=\frac{399}{198}$
Therefore,
$2.0 \overline{15}$ in $\frac{p}{q}$ form is $\frac{399}{198}$.