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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## A cirlce is drawn within the interior of a rectangle. Does tagged by: AAPL ##### This topic has 1 expert reply and 0 member replies ### Top Member ## A cirlce is drawn within the interior of a rectangle. Does ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult Manhattan Prep A circle is drawn within the interior of a rectangle. Does the circle occupy more than one-half of the rectangle's area? 1) The rectangle length is more than twice its width. 2) If the rectangle's length and width were each reduced by 25% and the circle unchanged, the circle would still fit into the interior of the new rectangle. OA D. ### GMAT/MBA Expert GMAT Instructor Joined 22 Aug 2016 Posted: 1909 messages Followed by: 30 members Upvotes: 470 AAPL wrote: Manhattan Prep A circle is drawn within the interior of a rectangle. Does the circle occupy more than one-half of the rectangle's area? 1) The rectangle length is more than twice its width. 2) If the rectangle's length and width were each reduced by 25% and the circle unchanged, the circle would still fit into the interior of the new rectangle. OA D. Given: A circle is drawn within the interior of a rectangle. Question: Does the circle occupy more than one-half of the rectangle's area? Let's take each statement one by one. 1) The rectangle length is more than twice its width. Note that if the length and the width of the rectangle are equal, i.e., the rectangle becomes a square; and the area of the biggest possible circle (a circle touching all the sides of the rectangle/square) would be less than the area of the rectangle. A rectangle with its length equal to twice its width is like two side-by-side squares. So, we have one biggest possible circle and two squares. It's obvious that the area of the biggest possible circle is less than half of the two squares. So, if the rectangle length is more than twice its width, the area of the biggest possible circle would be less than half of the rectangle. The answer is No, the circle does NOT occupy more than one-half of the rectangle's area. Sufficient. 2) If the rectangle's length and width were each reduced by 25% and the circle unchanged, the circle would still fit into the interior of the new rectangle. This is interesting... Since from Statement 1, we get the answer No; so if Statement 2 alone also suffices, we must get the answer No. Needless to say that for a very small circle, the area of the circle is less than half of the area of the rectangle. Let's see what happens when we have the biggest possible circle. Say the rectangle is a square and after the reduction of its length and the width, the circle (the biggest possible circle) touches the four sides. Say the diameter of the biggest possible circle is d; thus, its area = Ï€d^2/4 Since the circle touches the four sides, side of the square (after reduction) = d Thus, the side of the square before reduction = d/75% = 4d/3 Area of the original square (before reduction) = 4d/3 * 4d/3 = 16d^2/9 Ratio of (area of the biggest possible circle) to (area of the original square (before reduction)) = (Ï€d^2/4) / (16d^2/9) = (9Ï€) / 64 = ~27/64 < 1/2 Since half of 64 is 32, and the numerator is 27 (< 32), the area of the biggest possible circle is less than half of the rectangle. The answer is No, the circle does NOT occupy more than one-half of the rectangle's area. Sufficient. Even if you find out the actual value of 9Ï€, it will not less than 32. Note that 27 < 9Ï€ < 29. The correct answer: D Hope this helps! -Jay _________________ Manhattan Review GMAT Prep Locations: Manhattan Review India | Manhattan Review Hyderabad | Dilsukhnagar GMAT Courses | Mehdipatnam GRE Prep | and many more... Schedule your free consultation with an experienced GMAT Prep Advisor! Click here. • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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