The shaded region in the figure below represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the lenght and width of the frame, what is the length of the picture, in inches?
(A) 9*sqrt(2)
(B) 3/2
(C) 9/sqrt(2)
(D) 15(1-(1/2))
(E) 9/2
Total area = 18*15 = 270.
Since the frame and the picture have equal areas, the area of the picture = (1/2)270 = 135.
In the frame, L:W = 18:15 = 6:5.
The dimensions of the picture are in the same ratio:
Since 6:5 is close to 1:1, the length of the picture and the width of the picture are almost equal.
We can plug in the answers, which represent the length of the picture.
Since 12*12 = 144, the length of the picture must be around 12 inches.
Only
A works:
9√2 ≈ 9(1.4) = 12.6.
The correct answer is
A.
Algebraic solution:
Since L:W = 6:5, let L = 6x and the W = 5x.
Since L*W = 135, we get:
(6x)(5x) = 135
x² = 9/2
x = 3/√2.
The length of the picture = 6x = 6(3/√2) = 18/√2 * √2/√2 = (18√2)/2 = 9√2.
The correct answer is
A.
Last edited by
GMATGuruNY on Fri Apr 19, 2013 10:44 am, edited 2 times in total.
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