$\text{ Fringe width, }\beta =\frac{\lambda D}{d}$

where D is the distance between slits and screen and d is the distance between the slits. When D is doubled and d is reduced to half, then fringe width becomes

${\beta }^{\text{'}}=\frac{\lambda \left(2D\right)}{(d/2)}=\frac{4\lambda D}{d}=4\beta$