An object takes n times as much time to slide down a 45 ° rough...

$s = ut + \dfrac{1}{2}at^{2}, a = gsin\theta, u = 0$

Then, $s = 0 + \dfrac{1}{2}at^{2}$

or $t = \sqrt{\dfrac{2s}{a}} = \sqrt{\dfrac{2s}{gsin\theta}}$ [for smooth plane]

For the rough plane. the effective value of acceleration along the incline is
$a^{'} = gsin\theta - {μ}_{k}gcos\theta$

and $t^{'} = \sqrt{\dfrac{2s}{a}} = \sqrt{\dfrac{2s}{gsin\theta - {μ}_{k}gcos\theta}}$

Now, $\dfrac{t '}{t} = n = \sqrt{\dfrac{gsin\theta}{gsin\theta - {μ}_{k}gcos\theta}}$

$= \sqrt{\dfrac{\sin 45 °}{\sin 45 ° - {μ}_{k}\cos 45 °}}$

$= \sqrt{\dfrac{1}{1 - {μ}_{k}}}$

$⇒ \dfrac{1}{n2} = 1 - {μ}_{k} ⇒ {μ}_{k} = 1 - \dfrac{1}{n2}$