Hi,

The info and observations from the Q..
1) all numbers are >0.
2) there is no numeric value at all in the options/statements.

In the given circumstances, you cannot find the value. ANSWER at the best will in variables .

So E
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We need to find (x+z)/2

Statement 1. x-y = y-z . this means z, y, x are in Arithmetic Progression (increasing order also)..
And x+z = 2y or (x+z)/2 = y. But we don't know the value of y, so insufficient.

Statement 2. Clearly insufficient.

Combining the two statements: Let the three AP terms: z = z, y = z+d, x = z+2d
Now from second statement we are given that x^2 - y^2 = z
or (z+2d)^2 - (z+d)^2 = z
Solving we get 2zd + 3d^2 = z.. But this is NOT sufficient to find the value of z/d (thus that of y too) so the question cannot be answered.

Hence E answer

Clearly E since we have 3 variables and only 2 equations
moreover

1) x-y=y-z
y= (x+z) /2 we dont know x,y,z A,D gone

2) x^2-y^2 = z
(x-y) (x+y) =z. we dont know x,y,z B gone

put y = x+z /2 in 2nd equation
which will give quadratic but mean of x and z will be impossible to obtain... Insufficient C GONE

Remaining E is answer
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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.

Hi All,

We're told that X, Y and Z are POSITIVE numbers. We're asked for the average of X and Z. This question can be solved with a bit of math and TESTing VALUES.

1) X - Y = Y - Z

We can simplify this equation...
X + Z = 2Y

IF.... X=1, Z=1, Y=1, then the answer to the question is... (1+1)/2 = 1
IF.... X=2, Z=4, Y=3, then the answer to the question is... (2+4)/2 = 3
Fact 1 is INSUFFICIENT

2) X^2 - Y^2 = Z

IF.... X=2, Y=1, Z=3, then the answer to the question is... (2+3)/2 = 2.5
IF.... X=3, Y=1, Z=8, then the answer to the question is... (3+8)/2 = 5.5
Fact 2 is INSUFFICIENT

Combined, we can combine the two given equations....
X + Z = 2Y
X^2 - Y^2 = Z
into....
X + (X^2 - Y^2) = 2Y
X + X^2 = Y^2 + 2Y

Given that last equation - and since all 3 variables are POSITIVE - as X gets bigger, Y will ALSO get bigger. Since X could become a huge number (and Z is positive so it cannot 'offset' the increase in X'), the answer to the question "what is the average of X and Z?" will vary.
Combined, INSUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Hi lolmenot,

In your example (X = 10, Y = 8, Z = 6), the answer to the question is (10+6)/2 = 8. What if you used different values though? Would the answer always be 8 or could it be anything else? If the answer to the question changes, then Fact 1 is insufficient.

GMAT assassins aren't born, they're made,
Rich
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Hi Bunuel

Aren't the two Statements Contradictory

What One says is that y is the average of x&z
So we can say x<y<z or z<y<x

So if we look at statement 2 we have

(x+y)(x-y)=z

Since z is positive then x>y
we can infer that y<x<z

Where am i going wrong?
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1. Says that the difference b/w the x&y and y&z is same. this means they are in AP. But we wont get "One" value of y(it is the mean of x & z as per the stmt 1). Changing the values of X & Z, you get different values of Y.

2. Second stmt merely says that (x+y)(x-y) = z but we need answer for (X+Z)/2. hence this stmt is insufficient.

Combining the 2 stmts you get 2 equations . ie. from st 1. y = (x+z)/2 and from st 2 (x+y)(x-y) = z

but again note that there are 3 unknowns and 2 equations. Hence you will still not get "One" value for (x+z)/2

Hope this helps


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