Question

Express $2.4\overline{178}$ in the from $\frac{p}{q}$

Answer 1

Let $x=2.4178178$…. ………$\left(1\right)$

Here periodicity is $3$

So, we have to multiply x with $1000$

$\Rightarrow 1000x=1000\times 2.4178178$….

$\Rightarrow 1000x=2417.81781781$…… $\left(2\right)$

Now,

Equation $\left(2\right)$ - $\left(1\right)$ to,

$1000x-x=\mathrm{2417.81781781...}-2.4178178$

$\Rightarrow 999x=2415.4000$

$\Rightarrow x=\frac{2415.4000}{999}$

$\Rightarrow x=\frac{24154}{9990}$

Therefore, $\frac{p}{2}=\frac{24154}{9990}$

Hence, this is the answer.