A CIRCLE IS DRAWN WITHIN THE INTERIOR OF A RECTANGLE

[iamreadyforaction]

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.

The correct answer (d) works only if the circle is touching the sides of the rectangle (or the diameter of the circle is the maximum length possible given the size of the rectangle). My question is why the circle can not be a very small circle within the rectangle that does not touch the sides of the rectangle (in this case I believe the answer is E).

There is a similar CAT question seems to verify this possibility..
A cylindrical tank has a base with a circumference of 4sqrt(3pi) meters and an isosceles right triangle painted on the interior side of the base. A grain of sand is dropped into the tank, and has an equal probability of landing on any particular point on the base. If the probability of the grain of sand landing on the portion of the base outside the triangle is 3/4, what is the length of a side of the triangle?

On the answer to this question, the triangle was not limited to the constraints apparent in the first question (touching the sides). I realize that once a couple calculations are done for the second question, the triangle physically cannot encompass the maximum area of the base. However, my bolded question above does not seem to use this similar logic. Furthermore, the wording for the two questions are also quite similar.

[jnelson0612]

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.

The correct answer (d) works only if the circle is touching the sides of the rectangle (or the diameter of the circle is the maximum length possible given the size of the rectangle). My question is why the circle can not be a very small circle within the rectangle that does not touch the sides of the rectangle (in this case I believe the answer is E).

There is a similar CAT question seems to verify this possibility..
A cylindrical tank has a base with a circumference of 4sqrt(3pi) meters and an isosceles right triangle painted on the interior side of the base. A grain of sand is dropped into the tank, and has an equal probability of landing on any particular point on the base. If the probability of the grain of sand landing on the portion of the base outside the triangle is 3/4, what is the length of a side of the triangle?

On the answer to this question, the triangle was not limited to the constraints apparent in the first question (touching the sides). I realize that once a couple calculations are done for the second question, the triangle physically cannot encompass the maximum area of the base. However, my bolded question above does not seem to use this similar logic. Furthermore, the wording for the two questions are also quite similar.

Actually, the wording that the circle "is drawn within the interior of a rectangle" guarantees that the circle is NOT touching the sides of the rectangle. So it could be a very small circle within the rectangle, as you stated, or it could be one *almost* touching three sides of the rectangle. Using those clues, though, can the circle ever occupy more than one half of the triangle's area? Go back and see if you can figure out why the answer will be definitively NO using both clues, and please let us know if you need more help. :-)

[linzhouu]

Hi ,
I came across this question in CAT2.
I am not sure how statement 2 is sufficient.
Here is what solution states (followed by my question)
Let L and W stand for the length and width, respectively, of the original rectangle (whose area is LW). From this statement, then, the circle’s diameter is smaller than both 3/4L and 3/4 W (since the circle must fit into the interior both length-wise and width-wise). Because the diameter is twice as long as the radius, the radius must be smaller than either 3/8L or 3/8W .

Use those two dimensions in place of the two r’s in the area formula: The circle’s area = Ï€(less than 3/8L )(less than 3/8W ),

My Ques : the area calculated above is that of an elipse (pi*a*b) and not of circle .. the area of rectangle is a function of two variables, L and W , whereas that of the circle will be of one variable either L or W (Pi*3/8L*3/8L OR Pi*3/8W*3/8W) so we can not be sure if the area is less than 50% of rectangle or not .. so the answer to this should be A ...

Please explain if my solution is wrong.

[jlucero]

Hi ,
I came across this question in CAT2.
I am not sure how statement 2 is sufficient.
Here is what solution states (followed by my question)
Let L and W stand for the length and width, respectively, of the original rectangle (whose area is LW). From this statement, then, the circle’s diameter is smaller than both 3/4L and 3/4 W (since the circle must fit into the interior both length-wise and width-wise). Because the diameter is twice as long as the radius, the radius must be smaller than either 3/8L or 3/8W .

Use those two dimensions in place of the two r’s in the area formula: The circle’s area = Ï€(less than 3/8L )(less than 3/8W ),

My Ques : the area calculated above is that of an elipse (pi*a*b) and not of circle .. the area of rectangle is a function of two variables, L and W , whereas that of the circle will be of one variable either L or W (Pi*3/8L*3/8L OR Pi*3/8W*3/8W) so we can not be sure if the area is less than 50% of rectangle or not .. so the answer to this should be A ...

Please explain if my solution is wrong.

Your key point here is that rectangles are the function of two variables (l, w), while circles are a function of one (r). That's also the key to this question. Because one side length of the rectangle could be infinitely large and we would say that the circle takes up just a small fraction of that rectangle. But even if the circle were taking up the largest possible area of the modified, smaller rectangle, it would still be LESS THAN half the area of the original rectangle. Since you get a consistent answer here, statement 2 is sufficient.

[stephanie.y.liu]

I'm not sure I understand the explanation quoted from the CAT exam for statement 2. I also tried to compare the rectangle's area with pi*(3/8 L)^2 and pi*(3/8 W)^2. Why was the combination of the two used instead? pi*(3/8 L)(3/8 W)

[RonPurewal]

I'm not sure I understand the explanation quoted from the CAT exam for statement 2. I also tried to compare the rectangle's area with pi*(3/8 L)^2 and pi*(3/8 W)^2. Why was the combination of the two used instead? pi*(3/8 L)(3/8 W)

stephanie, your idea here is the right idea.

the point that the answer key is trying to convey -- in slightly uglier, but easier-to-understand, notation -- is that the area is
pi(thing)(thing)
where "thing" is less than both 3L/8 and 3W/8.

if this is unclear in the answer key, then i can submit the issue for editing.
thanks.

[jnelson0612]

:-)

[jonvindjohnsen]

It says in the explanation that the circle's diameter is less than the shorter dimension of the rectangle.

I'm wondering why it isn't less than OR equal to the shortest dimension?

If I picture 2 squares next to each other with each square a side of 1. Then max diameter would be 1, touching 3 sides, and still WITHIN the rectangle.
Of course the circle could be smaller, touching two sides or none. But why is the max. case not possible?

Thanks!

[RonPurewal]

The boundary of a region is not "within the interior of" that region.

[minamechem]

Thanks for sharing.

[RonPurewal]

Sure.

[lsyang1212]

Is it fair to say that when a question uses the term "rectangle", that this only means an actual rectangle with Length different from Width, or can "rectangle" also mean a Square, since all squares are technically also rectangles? Let me know if there's a flaw in my thinking.
Thank you!

[RonPurewal]

The term "rectangle" includes squares.

[sahilk47]

Hi Ron

I got the same question in my CAT exam today and I spent a lot of time to figure out the question and the approach to do the same. Barring the 2-2.30 minutes golden rule, how much time would you suggest one must take to solve such a question?

Thank you

[tim]

Your best bet is to forget you ever heard the words "two minutes" in relation to math problems. Each question will take a certain amount of time for you to do, and your job is to get good at predicting how long a given problem will take. Then if you think a problem will take too much time (more than 3 minutes, say), get rid of it as soon as you realize this. If you think you can do a problem in less than 2.5-3 minutes, go ahead and commit to finishing the problem and then only bail out if the problem ends up taking a lot longer than you estimated (because that means you probably don't know what you're doing after all).

There is no point in discussing how much time a problem "should" take, because that will be different for each student. The point is to make sure you are using timing strategies appropriate to your unique situation.

And remember, there is NOTHING that says you have to solve every GMAT problem in 2 minutes or even 2.5 minutes. As long as you finish all 37 questions in 75 minutes and are not wasting time or opportunities to use your time effectively, it doesn't matter how you got there.