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bblast: Legendary Member; Posts: 1077; Joined: Mon Dec 13, 2010 1:44 amFollowed by: 33 members; Thanked: 118 times; GMAT Score:710

by bblast » Tue May 31, 2011 9:15 am

The ratio of the area of a square mirror to its frame is 16 to 33. If the frame has a uniform width (c) around the mirror, which of the following could be the value, in inches, of c ?

I. 2
II. 7/2
III. 5

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ans 1,2 and 3

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Source: Beat The GMAT — Problem Solving | Tue May 31, 2011 9:15 am

cans: Legendary Member; Posts: 1309; Joined: Mon Apr 04, 2011 5:34 amFollowed by: 123 members; Location: India; Thanked: 310 times; GMAT Score:750

by cans » Tue May 31, 2011 9:24 am

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..

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Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Tue May 31, 2011 9:24 am

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.

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SoCan: Senior | Next Rank: 100 Posts; Posts: 97; Joined: Sun May 15, 2011 9:19 amFollowed by: 1 member; Thanked: 18 times

by SoCan » Tue May 31, 2011 9:41 am

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.

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