The ratio of the intensity at the centre of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the centre is

2

Two waves of a single source having an amplitude A interfere. The resulting amplitude

$A_r^2 = A_1^2 + A_2^2 + 2A_1A_2 \cos δ$

where $A_1 = A_2 = A$ and $δ$ = phase difference between the waves

$⇒ I_r = I_1 + I_2 + 2 \sqrt{I_1I_2}\cos δ$

When the maxima occurs at the center, $δ = 0$

$⇒ I_{r_1}= 4I$ …(1)

Since the phase difference between two successive fringes is 2π, the phase difference between two points separated by a distance equal to one quarter of the distance between the two, successive fringes is equal to $δ = (2π)(\frac{1}{4})=\frac{π}{2}$ radius

$⇒ I_{r_2}=4I \cos^2(\frac{π/2}{2})= 2I$ …(2)

using (1) and (2)

$\frac{I_{r_1}}{I_{r_2}}=\frac{4I}{2I}=2$