Express each of the following decimals in the form $\frac{p}{q}$:$ 0 . \overline{54} $

Given:

The given decimal number is \( 0 . \overline{54} \).

To do:

We have to express the given decimal in $\frac{p}{q}$ form.

Solution:

$0. \overline{54}$

Let $x = 0.5454....$

Multiply both sides by 10.

$10x = 10(0.5454....)$

$10x = 5.454......$

Multiply both sides by 100.

$100x = 100(0.5454....)$

$100x = 54.5454.....$

Therefore,

$100x-x = 54.5454.... - 0.5454.....$

$99x = 54$

$x = \frac{54}{99}$

$x=\frac{9\times6}{9\times11}$

$x=\frac{6}{11}$

Therefore,

$0. \overline{54}$ in $\frac{p}{q}$ form is $\frac{6}{11}$.  

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Updated on: 10-Oct-2022

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