Statement (1) This tells us A and B are similar rectangles, in which case the ratio of their areas is the square of the ratio of two corresponding single dimensions, such as the two diagonals. This statement is sufficient.
(Proof: By the Pythagorean theorem, 102 = LA2 + (kLA)2 and 202 = LB2 + (kLB)2, where LA and LB are lengths of A and B, and k is the ratio of width to length. We then have 100 = LA2(1+k)2 and 400 = LB2(1+k)2. Isolating (1+k)2 in both equations gives us the equality LA2/LB2 = 1/4 = LA2k/LB2k, which is the ratio of the areas.)
The correct answer must be either A or D.
Statement (2) Since the ratio of the diagonals and the ratio of the widths are both 1/2, A and B are similar. This statement is sufficient.
(Proof: By the Pythagorean theorem, we have W2 + LA2 = 100 and (2W)2 + LB2 = 400. Isolating (100 − W2) in both equations gives us the equality LA2/LB2 = 1/4.)
The answer is D.