Hello,
Can you please tell me how to solve this:
The figure above is a square divided into rectangles. The measures of some of the
sides of some of the rectangles are shown in the figure. What is the value of
x if the shaded area equals the unshaded area?
(A) 0.5
(B) 1
(C) 1.5
(D) 3
(E) 4.5
OA: C
Thanks a lot,
Sri
Square divided into rectangles
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Hi Sri,
The key to solving this question is that big shape is a SQUARE.
From the drawing, we know that we're dealing with a 9x9 square (the right "side" of the square is 3 3's, so we have a length of 9).
The first "column" of rectangles has a length of X
The third "column" of rectangles has a length of 3
The second "column" of rectangles is unlabeled, so let's say is has a length of Y
We're told that the total area of the shaded region = the total area of the unshaded region.
The area of the 5 shaded regions = 3X + 9 + 3Y + 3X + 9
The area of the 4 non-shaded regions = 3Y + 3X + 9 + 3Y
Now, we set them equal to one another...
6X + 3Y + 18 = 3X + 6Y + 9
Combine like terms...
3X + 9 = 3Y
X + 3 = Y
Since the square is a 9x9 square, we know that X+Y+3 = 9; this means....
X + Y = 6
Now we have a system of equations:
X+3 = Y
X+Y = 6
X=1.5 and Y=4.5
We're asked for the value of X:
Final Answer: C
GMAT assassins aren't born, they're made,
Rich
The key to solving this question is that big shape is a SQUARE.
From the drawing, we know that we're dealing with a 9x9 square (the right "side" of the square is 3 3's, so we have a length of 9).
The first "column" of rectangles has a length of X
The third "column" of rectangles has a length of 3
The second "column" of rectangles is unlabeled, so let's say is has a length of Y
We're told that the total area of the shaded region = the total area of the unshaded region.
The area of the 5 shaded regions = 3X + 9 + 3Y + 3X + 9
The area of the 4 non-shaded regions = 3Y + 3X + 9 + 3Y
Now, we set them equal to one another...
6X + 3Y + 18 = 3X + 6Y + 9
Combine like terms...
3X + 9 = 3Y
X + 3 = Y
Since the square is a 9x9 square, we know that X+Y+3 = 9; this means....
X + Y = 6
Now we have a system of equations:
X+3 = Y
X+Y = 6
X=1.5 and Y=4.5
We're asked for the value of X:
Final Answer: C
GMAT assassins aren't born, they're made,
Rich
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Since the height of each row is 3:
All 3 A regions have the same area (3x).
All 3 B regions have the same area (3y).
All 3 C regions have the same area (3*3 = 9).
Sum of the shaded regions = 2A + B + 2C.
Sum of the unshaded regions = A + 2B + C.
Since the sums are equal, we get:
2A + B + 2C = A + 2B + C
A + C = B.
Since A=3x, C=9, and B=3y, we get:
3x + 9 = 3y
x+3 = y.
x-y = -3.
Since the garden is square, each side = 9.
Thus, x+y=6, as shown in the figure above.
Adding the two equations, we get:
(x-y) + (x+y) = -3+6
2x = 3
x = 1.5.
The correct answer is C.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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