In Young's double-slit experiment, fringe width $\beta = \dfrac{\lambda D}{d}$. If the screen distance $D$ is doubled and the slit separation $d$ is halved, the new fringe width is:

Plug in directly. New $\beta' = \dfrac{\lambda \cdot (2D)}{(d/2)} = \dfrac{2 \lambda D}{d/2} = 4 \cdot \dfrac{\lambda D}{d} = 4\beta$. Intuition: increasing $D$ stretches the geometry linearly, and decreasing $d$ shrinks the angular separation linearly. Both effects widen the fringes, and together they multiply.