Manhattan GMAT Challenge Problem of the Week – 15 June 2010

by Manhattan Prep, Jun 15, 2010

This is our latest Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go! (And don't forget to answer the Showdown for a chance to win our Strategy Guides.)

Question

In the xy-plane, region Q consists of all points (x, y) such that [pmath]x^2 + y^2[/pmath] 100. Is the point (a, b) in region Q?

(1) a + b = 14

(2) a > b

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.

Answer

The first step is to figure out what region Q represents. Lets consider the boundary of region Q by ignoring the less than part: [pmath]x^2 + y^2[/pmath] = 100. This equation represents a circle in the xy-plane, centered on the origin, with a radius of 10 (the square root of 100). Thus, region Q consists of all points on or inside this circle. We are asked whether point (a, b) lies inside this region. We can rephrase the question by substituting a for x and b for y: do the variables a and b always satisfy the inequality [pmath]a^2 + b^2[/pmath] 100?

Statement (1): INSUFFICIENT. Relatively quickly, we can find a point on the line a + b = 14 that does not satisfy the inequality. Choose a = 0. Then b = 14, and the sum of the squares is 196, which is greater than 100. Thus, in this case, (a, b) would not fall within region Q.

However, can we find any point on or within the circle? If we make both a and b equal 7, then the sum of their squares is 49 + 49 = 98, which is less than 100. We could also choose a = 8 and b = 6, which gives us the sum of squares 64 + 36 = 100. Either case satisfies the inequality, and so (a, b) in these cases would fall within region Q.

Statement (2): INSUFFICIENT. The condition that a is greater than b is not very restrictive. We can find points that meet this condition both inside and outside the circle. For instance, (1, 0) is within the circle, but (101, 100) is not.

Statements (1) and (2) together: INSUFFICIENT. The case from statement 1 in which both a and b equaled 7 is no longer valid, but the case of a = 8 and b = 6 still works. Thus, we have a point satisfying both statements that lies on the circle. In fact, some suitable points lie within the circle, such as (7.5, 6.5). However, we can still find suitable points that lie outside the circlefor instance, (14, 0).

The correct answer is (E).

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