The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:

First, draw the coordinate plane and add in point A, which has coordinates (d,d). Since d is greater than 0, this point lies in the first quadrant, as shown:

Notice that the origin, the point (0, d), the pint (d, 0), and point A (d,d) form the corners of a square:

Now, we draw the line segment between (0,0) and (d,d). This is the diameter of a circle, which we also draw:

Notice that the answer doe not depend on d. All that matters is that the circle contains a square. The fraction of the circle’s area in the first quadrant will be the same, no matter what. Thus, we can drop d as a variable and create a new variable r for the radius of the circle. It will be easier to compute areas in terms of r (which will also cancel out in the end).

The area of the circle is just ?r².

The fraction we are looking for is this:

So we need the shaded area of the circle, which consists of the square ( ) and 2 of the four “leaves of the table” (  ).

The square’s area can be found this way:

Now the area of the 4 leaves (  ) is the area of the circle (  ) minus the area of the square ( ). So the area of the 4 leaves is ?r² – 2r² = (? – 2)r².

This means that the area of 2 leaves is .

The whole shaded area is thus .

Finally, the desired fraction is .

The correct answer is D.