Is the area of a certain square greater than the area of a certain rectangle? (1) A side of the rectangle equals to a side of the square. (2) A side of the rectangle is twice longer than a side of the square.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) The area of a square is a², where a is the side of the square. The area of a rectangle is bc, where b and c are the sides of the rectangle.

Statement (1) tells us that the figures have equal sides. Clearly, that is not enough. If the shorter side of the rectangle is equal to the side of the square, then the area of the rectangle is greater. If the longer side of the rectangle is equal to the side of the square, then the area of the square is greater. If the rectangle is a square as well, then their areas are equal. Therefore statement (1) by itself is NOT sufficient.

Statement (2) tells us that a side of the rectangle is twice longer than a side of the square. If the other side of the rectangle is longer as well, then the area of the rectangle is greater. However if the other side of the rectangle is shorter than half the side of the square, then the area of the square is greater. Statement (2) by itself is NOT sufficient.

If we use the both statements together the area of the rectangle is greater, because in this case the sides of the rectangle must be a and 2a. So the area of the rectangle is 2a², which is greater than the area of the square, a². The correct answer is C. ----------

I'm confused why this isn't E?

I would of thought that considering we can't determine that statement 1 & 2 are relating to the same side or not, we can't determine if the rectangle is bigger/smaller?

e.g. If the two statements were for different sides of the rectangle, I agree that statement C is correct.

BUT if there are the same side, then the other side could be .0000001 or 10,000,000 and thus it could be smaller or larger.

If the two statements were for different sides of the rectangle, I agree that statement C is correct.

The statements refer to different sides of the rectangle, when combined, because just one side of the rectangle can not be equal and be twice longer than the side of the square AT THE SAME TIME.

i.e. The system b = a b = 2a is NOT possible for positive a and b (it's possible only if a = b = 0).

Let's take a look at the both statements combined once again. (1) A side of the rectangle equals to a side of the square. This statement transforms into b = a (we chose side b, but could have chosen side c. Considering the denotation, it does NOT affect the reasoning).

(2) A side of the rectangle is twice greater than a side of the square. We already know that b = a, so c = 2a. Note, that since b = a, then side c is twice longer than side b in the rectangle. So c is the larger side of the rectangle.

The areas are a² vs. bc = a × (2a) = 2a². So the area of the rectangle is greater.

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