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 $\frac{399}{198}$.

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Updated on: 10-Oct-2022

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