Data Sufficiency

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's 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.

Great question. It is often helpful in data sufficiency to paraphrase the question stimulus in your own words, as it helps to make sure that you really understand what the question is asking (just be careful you don't change the ask!). Here, asking if a circle occupies more than half of the rectangle's area, because the circle is inside of it, is just asking "Is the area of the circle over 1/2 as big as the area of the rectangle?"

Statement 1: A circle's diameter cannot be larger than the shortest side of a rectangle, so its radius cannot be larger than 1/2 of the shortest side. Because the width is less than 1/2 of the length, this means that the radius of the circle is at most 1/2 of the width. If you think spatially about this, you can visualize that over half of the rectangle will have no part of the circle at all (because length is over twice as big as width), and that there will still be some space even in the portion that the circle is in (think about the shapes of a circle and rectangle, there will be some white space between the circle and rectangle). Drawing a rough sketch can help with this visualization. You can also pick numbers, so say length is 8 and width is 4 (keeping in mind length would have to be more than 8, but could be 8.000001). Radius is then at biggest 2. Rectangle area is 8*4 = 32. Circle area is at most 3.14*2*2 = ~12.5. Not even close to half. SUFFICIENT.

Statement 2: This one in my opinion is slightly trickier, and best solved picking numbers. Pick small numbers that can easily be reduced by 25%. Also, make the rectangle a square, because a square is just a special type of rectangle and this will maximize the circle's area relative to the rectangle (because it is limited by the shortest side). We already know that of course it could be less (it could be a tiny dot in the middle of a huge rectangle), so we are trying to test if it can be greater than 1/2 of not. Picking 4 by 4 for the square, area = 16. Reduced by 25% gives us 3 by 3, area now = 9. Maximize the circle, means diameter = 3 and radius = 3/2. 3.14*3/2*3/2 = 3.14*9/4 = roughly 28/4 = 7. To estimate 3.14*9, I just multiplied 9*3 to get to 27 and then realized that .14*9 will just be a little over 1, which gives me a little over 28. Now I can see that 7 (or even a tad over) is going to be less than 16, so there is no way that this circle is over 1/2 the area of the rectangle. SUFFICIENT.

Hence answer D, both statements are sufficient alone.

I HOPE THIS HELPS!!!