Final answer:
Halving the separation between the slits as well as the distance from the slits to the screen does not change the fringe width; the fringe width depends directly on the ratio of these two distances and the wavelength.
Step-by-step explanation:
When both the separation between the slits and the distance from the slits to the screen are halved, the fringe width in a double-slit interference pattern remains unchanged. According to the formula Ay = xλ/d, where Ay is the fringe width, x is the distance to the screen, λ is the wavelength, and d is the slit separation, by halving both x and d, the ratio x/d stays the same. Thus, the fringe width does not change because it depends directly on the ratio of x to d and the wavelength λ, which is considered constant in this context.