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Express each of the following decimals in the form $\frac{p}{q}$:
\( 0.4 \overline{7} \)
Given:
The given decimal number is \( 0 . 4\overline{7} \).
To do:
We have to express the given decimal in $\frac{p}{q}$ form.
Solution:
$0. 4\overline{7}$
Let $x = 0.47777....$
Multiply both sides by 10.
$10x = 10(0.47777....)$
$10x = 4.7777.....$
Multiply both sides by 100.
$100x = 100(0.47777....)$
$100x = 47.7777.....$
Therefore,
$100x-10x = 47.7777.... - 4.7777.....$
$90x = 43$
$x = \frac{43}{90}$
Therefore,
$0. 4\overline{7}$ in $\frac{p}{q}$ form is $\frac{43}{90}$.
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