Statement (1) can be simplified and looked at conceptually. No need to pick numbers.
(x + z)/2 = y
Since we have no idea what y is, statement (1) is insufficient.
Statement (2) tempts us to use the difference of squares common equation.
(x + y)(x - y) = z
Since x, y and z are all positive, we know x - y must be positive, or x - y > 0. So x > y. But this is about all we can get with this approach, and this is something we could have observed from x^2 - y^2 = z, since z is positive.
The fact that there are no values in x^2 - y^2 = z is a clue that there are many possible combinations of x, y and z that could work. In picking numbers, I always like to start with simple numbers that fit the situation. So since x > y, here are some numbers that fit:
x = 2
y = 1
z = 2^2 - 1 = 3
(x + z)/2 = 2.5
OR
x = 3
y = 2
z = 3^2 - 2^2 = 5
(x + z)/2 = 4
Since we get two possible values for (x + z)/2, statement (2) is insufficient.
For (1) and (2) together, here's an alternative to picking numbers. Notice that (1) tells us that y is the midpoint of x and z. Since (2) tells us that x > y, we know that z < y < x, and they are all evenly spaced. I like the idea of representing the spacing as d, so y = z + d, and x = z + 2d. Therefore, (2) gives us:
(z + 2d)^2 - (z + d)^2 = z
z^2 + 4dz + 4d^2 - z^2 - 2dz - d^2 = z
2dz + 3d^2 = z
3d^2 = z - 2dz = z(1-2d)
z = 3d^2/(1-2d)
In order to keep z positive, we must have d < ½, but as long as we do that, z could be lots of numbers, and therefore y and x could also be lots of numbers. Since (1) showed that our question is equivalent to y = ?, (1) and (2) together are insufficient.
Stuart Park
Simply Brilliant
Stuart is a Harvard grad GMAT expert who scored 760 the first time he took the exam, with 99th percentile quant and verbal scores. He has extensive experience teaching for one of the "elite" GMAT prep companies. Through
https://www.simplybrilliantprep.com he offers online classes, private tutoring and MBA application consulting for clients worldwide.
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