let side of square = a, thus area of square mirror = a^2
Also frame area = total area - mirror area = (a+2c)^2 - a^2 = 4c^2+4ac
thus ration = a^2/(4c^2+4ac) = 16/33 (this is given)
cross multiply:- 33a^2 = 64c^2+64ac
or 33a^2 -64ac - 64c^2 = 0
if we solve this, we will get c in terms of a.
As a is not given, c can take any value..

Hi,
ar(inner square) /ar(frame) = 16/33
So, ar(inner square) /[ar(outer square) - ar(inner square)] = 16/33
So ar(inner square) /ar(outer square) = 16/16+33 = 16/49
So side length of inner square/side length of outer square = 4/7
If 4x is inner length, c = 3x.

Cheers!

While it's good to think about the equations while you're studying, you'll save a lot of time during the test if you think about questions like this one conceptually.

There are no constraints on what the areas of the mirror or its frame could be besides their ratio. Therefore, you can set c to whatever value you'd like. Once you choose c, the rest of the dimensions are obviously constrained. It's like asking, "if a rectangle's longer side is twice as long as its shorter side, how long can the shorter side be?" Anything, but once you choose, the longer side is constrained. Thinking about the question like this should allow you to solve it in 10-20 seconds, freeing up valuable time for questions that require more work.

Of course, the question would be trickier and would require more work if there was another constraint placed - e.g., the area of the mirror must be an integer, the side of the frame must be a multiple of 3, etc.