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GMAT Prep 3 Qn

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GMAT Prep 3 Qn [#permalink] New post 15 Nov 2013, 11:00
Guys these are the questions from the new GMAT prep . Please help me solve. :?:

Q1. For a display, identical cube boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom layer has 81 boxes and the top layer has only 1 box, how many boxes are there in display ?
236
260
269
276
285

Q2. If a rectangular region has perimeter P inches and area A square inches , is the region square ?
(1)P = 4/3 * A
(2) P =4 * A^1/2



\\
OA1 : E and OA2 : B
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Re: GMAT Prep 3 Qn [#permalink] New post 15 Nov 2013, 17:13
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ankitongmat wrote:
Guys these are the questions from the new GMAT prep . Please help me solve. :?:

Dear ankitongmat,
I'm happy to help. :-) In the future, I will ask you to post only one question per post.

Q1. For a display, identical cube boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom layer has 81 boxes and the top layer has only 1 box, how many boxes are there in display ?
236
260
269
276
285


The first row has 9 x 9 = 81 cubes; the second row has 8 x 8 = 64 cubes; the third row has 7 x 7 - 49 cubes, and so forth.
Total = 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1
let's make the arithmetic easy:
Total = (64 + 36) + (25 + 16 + 9) + (49 + 1) + 81 + 4 = 100 + 50 + 50 + 81 + 4 = 285
Answer = (E)

Q2. If a rectangular region has perimeter P inches and area A square inches , is the region square ?
(1) P = 4/3 * A
(2) P = 4 * A^1/2


Let's say the rectangular region x by y.
Area = x*y
Perimeter = 2x + 2y
The question is: does x = y?

(1) P = 4/3 * A
3P = 4A
3(2x + 2y) = 4xy
6x + 6y = 4xy
3x + 3y = 2xy
This is one equation with two unknowns. If we required x = y, we could solve for a unique solution. If we required, say x = 2y, we would solve for another unique solution. There's no reason to wade through all that algebra. It's enough to see that different requirement lead to different solutions, and there is nothing from this that requires that x = y. Therefore, this statement, alone and by itself, is insufficient.

(2) P = 4 * A^1/2
Square both sides, to get rid of the radical
2x + 2y = 4[(xy)^(1/2)]
Divide everything by 2
x + y = 2[(xy)^(1/2)]
Now, square both sides
(x + y)^2 = 4xy
For the left term, squaring the quantity (x + y) see this blog:
http://magoosh.com/gmat/2013/three-alge ... -the-gmat/
(x^2) + 2xy + (y^2) = 4xy
Subtract 4xy from both sides:
(x^2) - 2xy + (y^2) = 0
This is another important algebra equation from that same blog, and it simplifies to:
(x - y)^2 = 0
Take a square root
x - y = 0
x = y
So this statement absolutely requires that x = y, which means that the rectangular region is indeed a square. Therefore, this statement, alone and by itself, is sufficient.
Answer = (B)

BTW, here's a blog with more "is it a square?" DS questions.
http://magoosh.com/gmat/2012/gmat-geome ... -a-square/

Does all this make sense?
Mike :-)
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Mike McGarry
Magoosh Test Prep

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Re: GMAT Prep 3 Qn [#permalink] New post 16 Nov 2013, 10:34
Dear Mike

Thanks for the solutions.
But in Q2 isn't there any alternative way to solve the question without taking x and y. I mean in exam how can we determine to use algebra in a simple looking geometry question like this. Does this question give any hint as to use algebra somewhere ? I marked my answer as D since through both of the options I concluded that it not a square since to make it a square I need P = 4A which is not there in any of the equations. Please let me know if any other alternatives are there to solve the question.

Ankit
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Re: GMAT Prep 3 Qn [#permalink] New post 17 Nov 2013, 16:04
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ankitongmat wrote:
Dear Mike

Thanks for the solutions.
But in Q2 isn't there any alternative way to solve the question without taking x and y. I mean in exam how can we determine to use algebra in a simple looking geometry question like this. Does this question give any hint as to use algebra somewhere ? I marked my answer as D since through both of the options I concluded that it not a square since to make it a square I need P = 4A which is not there in any of the equations. Please let me know if any other alternatives are there to solve the question.
Ankit

Dear Ankit,

My first thought was to use algebra. The fact that the second equation has a variable raised to a power --- right there, I would call that an invitation to do algebra, because aside from the Pythagorean Theorem, almost nothing in pure geometry involves raising anything to any powers.

Actually, now that I look at it again, I see statement #2 could be handled simpler. Notice that your equation, P = 4A, is incorrect. Suppose a square has a side of s. Then A = (s^2), but the P does not equal 4A --- it equals 4s. Therefore, s = A^(1/2), and P = 4*A^(1/2). That is precisely what statement #2 says. That's a much shorter and more elegant approach to statement #2.

Also, notice if you are looking for a particular algebraic condition, and it is not met, that may not mean a definitive "No" answer (which would be sufficient) ----- more often it means that something could be correct by doesn't have to be (that's not sufficient). That's precisely what is happening with statement #1.

Does all this make sense?
Mike :-)
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Mike McGarry
Magoosh Test Prep

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Re: GMAT Prep 3 Qn   [#permalink] 17 Nov 2013, 16:04
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