Intensity at centre of bright fringe,

$I_0=I+I+2\sqrt{II}·\cos 0$

$I_0=2I+2I$    $\left[\because \cos 0°=1\right]$

$=4I$

Similarly Intensity at point $\frac{P}{4}$

(with phase difference $=\frac{2\pi }{4}=\frac{\pi }{2}$)

$I^{\text{'}}=I+I+2\sqrt{II}\cos \frac{\pi }{2}$

$=2I+2\sqrt{II}\times 0$

$=2I$

$\therefore \frac{I_0}{I^{\text{'}}}=\frac{4I}{2I}=2$