Diffraction Grating Relationship Question

[Callix]

Homework Statement

A diffracting grating casts a pattern on a screen located a distance L from the grating. The central bright fringe falls directly in the center of the screen. For the highest-order bright fringe that hits the screen, m=x, and this fringe hits exactly on the screen edge. This means that 2x+1 bright fringers are visible on the screen. What happens to the number of bright fringes on the screen,

a). If the wavelnegth of the light passing through the grating is doubled
b). If the spacing d between adjacent slits is doubled

Homework Equations

Listed in attempt

The Attempt at a Solution

Using Young's Slits equations:
$\huge y_\text{bright}=\frac{\lambda L}{d}m \implies m=\frac{y_\text{bright}d}{\lambda L}$

a). m would halve
b). m would double

If someone could please check this, I would greatly appreciate it! I would also appreciate an explanation and steps to the right solution if this is wrong.

Thanks in advance!

 

[Callix]

Or perhaps since..

# of fringes = 2x+1, where x=m, then 2m+1

a). if m halves, then # of fringes = x+1
b). if m doubles, then # of fringes = 4x + 1

Or maybe I'm just overthinking it?