Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
We are told that both x and y are positive, and we are asked whether x is greater than 4. There’s not much rephrasing we can do to this question, so let’s 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 x2 + y2 > 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 we’ll 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. Let’s 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:
a) y is positive (from the stem).
b) y is less than 4 (that is, less than x).
c) x2 + y2 > 25.
Let’s plug 4 in for x in the last inequality. We get
16 + y2 > 25
y2 > 9
y > 3 (since y must be positive, we don’t 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.