ABCD is a square with a side y, and JKLM is a side x.

Question

square in a square.png ABCD is a square with a side y, and JKLM is a side x. If Rectangle S (not shown) with length (x + y) has the same area as the shaded region above, what is the width of Rectangle S? (A) x (B) y (C) y + x (D) y - x (E) y^2 - x^2 For a complete solution, see this post: https://magoosh.com/gmat/2013/three-alge ... -the-gmat/

GMATtracted · Answer

We have the area of the shaded region:-(y^2)-(x^2)
Now the shaded area can be factorised as (y+x)(y-x).
The question tells us that a rectangle of length (x+y) has the same area (y^2-x^2).
Hence, the width of the rectangle has to be (y-x)

manishuol · Answer

Hi Mike,

Sorry for Question ....but do you really think ?? This is a 700- level Question ??? I did this in around 30 seconds .......... Pls advise. Thanks !! in Advance ............

mikemcgarry · Answer

To be perfectly honest, many of the questions I post are ones I have recently written. I like to share brand new content with folks who are studying for the GMAT. If a question has been in the Magoosh product for a while, then hundreds of people will have answered it, and we can see the percentage correct and thus judge with considerable accuracy the difficulty of the question. By contrast, when I write a brand new question and post it on GC, the system requires me to estimate a difficulty level, and I have to take my best guess. To be perfectly blunt, I am conversant in multivariable calculus and advanced statistics, so pretty much all GMAT math looks easy to me ---- it's very hard for me to guess what other folks are going to find easy or difficult. I am estimating from my experience of the wide array of GMAT students I have encountered. With this particular question, one has to see, first of all, that the shaded area is the big square minus the small square, y^2 - x^2. Then, one has to see one can factor y^2 - x^2 = (y + x)(y - x). If one sees both of those right away, this problem is trivially easy. I would say, though, not all GMAT test takers will have both of those observations right at their fingertips. For some folks, either one or both of those will be completely befuddling. It's always the case, in any GMAT math problem --- if you see all the things that are necessary to see, the problem becomes quite easy. All of math has the quality that it's impossibly difficulty when you don't know what to do and trivially easy when you do know what to do. Does all this make this question an 700 question? I don't know, but I will say, I believe it says good things about your mathematical abilities that you were able to solve it so easily. Congratulations. Mike :-)

Bluelagoon · Answer

Just think on the lines of the question. You have to find the other side and you are given the area of the rectangle.
Area = L x B
L = x+ y
Area = y^2 - x^2
Hence B = y -x.

manishuol · Answer

Thanks !! for your Detailed Genuine reply ... Mike !! I appreciate that. :-D

Narenn · Answer

100% agree with the highlighted portion. It is because, these things are fitted in our brain so firmly that whenever we see y^2 - x^2 , we quickly go on to factories it as (y+x)(y-x). but this may not so easy for some students. By the way, the equation (y+x)(y-x) = (x+y)W would have become a dangerous trap in if y and x had not been the lengths. In that case we would not be able to divide both sides of the equation by (x+y) without knowing the exact values of x and y. Consider if y=-x I initially considered this possibility assuming that Mike sir will not give us such an easy task. :wink:

Regards, Narenn

mikemcgarry · Answer

You are quite welcome. Mike :-)

WpackAlumDukeBound · Answer

Agree with everything Mike said above. My brain didn't even jump to factoring y^2-x^2 , it wanted to plug numbers in right away. I set y=4 and x=1 y+x=5 Area of shaded region equals 15 15 = 5 x (z) z = 3 = y-x, Answer D MUCH longer than factoring y^2-x^2 out, but I still got there in under 2 minutes.

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ABCD is a square with a side y, and JKLM is a side x.

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ABCD is a square with a side y, and JKLM is a side x.

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