Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people that enter our challenge, the better the prizes!
Is xy an integer?
(1) x is the ratio of the area of a square to the area of the largest possible circle inscribed within that square.
(2) y is the ratio of the area of a circle to the area of the largest possible square inscribed within that circle.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
The question asks whether a particular product (xy) is an integer. Note that this is a Yes-No question.
Statement 1: NOT SUFFICIENT. This statement only refers to x. Without knowing anything about y, you cannot know whether xy is an integer.
Statement 2: NOT SUFFICIENT. Likewise, this statement only refers to y, so it cannot be sufficient.
Statements 1 and 2 TOGETHER: SUFFICIENT. The super-fancy way to get the answer is to realize that x is a fixed number, completely determined by its definition in the statements. The same is true of y. Why is this the case? All circles are the same shape, so they are all similar to each other. Likewise, all squares are the same shape and are similar to each other. So when you inscribe a circle inside a square (to touch all four sides of the square), there
There should be only one “shape” to the picture in your mind of a square with a circle inscribed inside it, touching all four walls. All that’s different is how large or small that picture is, so the ratio of the square’s area to the circle’s area is fixed:
The same is true for y, the ratio of the circle’s area to the area of an inscribed square:
So x and y are fixed. You don’t know what their values are, but you don’t care: in theory, you could calculate those values. And then you could determine whether the product is an integer or not.
The longer way to get the answer is to actually figure out these ratios.
Take x first. Call the side of the square 1. Then the radius of the inscribed circle is , and the area of the circle is . The area of the square is , so the ratio of the square’s area to the circle’s area is 1 : , or . That’s the value of x.
Now take y. Call the side of the square 1 again. Then the diameter of the circle is the diagonal of the square, which is . The radius of the circle is , and the area of the circle is = = . That’s the value of y, since the area of the square is just 1, and you want the ratio of the circle’s area to the square’s area.
Finally, the product of x and y is , which is indeed an integer.
The correct answer is C.
Special Announcement: If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner receives a number of our our Strategy Guides. The more people enter, the better the prize. Provided the winner gives consent, we will post his or her name on our Facebook page.