Question

Convert $12.3454545\dots$ into a rational number in $\frac{p}{q}$ form.

Answer 1

Let $x=12.34545\dots$
Here, The repeating digits of the given decimal fraction are ‘$45$’. So, we need to transfer repeating digits to the left of the decimal point. To do so, we need to multiply the original number by $1000$.
So, $1000x=12345.4545\dots eq\left(i\right)$
Now we have to shift the repeating digits to the right of the decimal point. To do so we have to multiply the original number by $10$.
$10x=123.4545\dots eq\left(ii\right)$
Subtracting $eq\left(ii\right)$ from $eq\left(i\right)$ we get,
⟹ $1000x–10x=12345.4545–123.4545$
⟹ $990x=12222$
⟹$x=\frac{12222}{990}=\frac{679}{55}$
Hence, the required rational fraction is $\frac{679}{55}$.