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Manhattan GMAT Challenge Problem of the Week  19 April 2011
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Question
If x and y are both positive, is x greater than 4?1) x > y
2) In the coordinate plane, the point (x, y) lies outside a circle of radius 5 centered at the origin.
(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
We are told that both x and y are positive, and we are asked whether x is greater than 4. Theres not much rephrasing we can do to this question, so lets move on to the statements.
Statement (1): INSUFFICIENT. All we know here is that x is larger than y. It is still possible for x to be any positive number (y just has to be a smaller positive number).
Statement (2): INSUFFICIENT. If the point (x, y) lies outside a circle of radius 5 centered at the origin, then that point lies at a distance of more than 5 units away from (0, 0). In the coordinate plane, distance can be computed with the Pythagorean Theorem. We can rephrase this statement to say that [pmath]x^2[/pmath] + [pmath]y^2[/pmath] > 25. However, x can still be as small or as large a positive number as we wish.
Statement (1) and (2) together: INSUFFICIENT. We can definitely pick a very large value for x to satisfy the statements and answer the question with a Yes. Just make y smaller to make the first statement true, and if x is bigger than 5, then well definitely have statement (2) true as well.
The trick is that we can pick a value of x that is not greater than 4 and still satisfy all the conditions. Lets try picking x equal to 4 (this would give us an answer of No to the question). So we need to see whether there are any values of y that satisfy the following 3 conditions:
 y is positive (from the stem).
 y is less than 4 (that is, less than x).
 [pmath]x^2[/pmath] + [pmath]y^2[/pmath] > 25.
Lets plug 4 in for x in the last inequality. We get
16 + [pmath]y^2[/pmath] > 25
[pmath]y^2[/pmath] > 9
y > 3
(since y must be positive, we dont have to worry about the negative possibilities)
The conditions become these: y is greater than 3 and less than 4. Any number between 3 and 4 satisfies the conditions. Notice that y is not restricted to integer values; nothing in the problem indicates that it should be. Thus, we still cannot definitively answer the question of whether x is greater than 4. There are other values of x less than 4 that will work; one tricky part of this problem is that those values of x are greater than 3.
The correct answer is E.
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