AbeNeedsAnswers wrote:Is the average (arithmetic mean) of the numbers x, y, and z greater than z ?

(1) z − x < y − z
(2) x < z < y

A

Source: Official Guide 2020

Target question: Is the average (arithmetic mean) of the numbers x, y, and z greater than z?
This is a good candidate for rephrasing the target question.
Rewrite the question as "Is (x + y + z)/3 > z?"
Multiply both sides by 3 to get: "Is x + y + z > 3z?"
Subtract z from both sides to get: "Is x + y > 2z?"
REPHRASED target question: Is 2z less than x + y?

Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: z − x < y − z
Add z to both sides to get: 2z − x < y
Add x to both sides to get: 2z < x + y
PERFECT!
The answer to the REPHRASED target question is YES, 2z IS less than x+y
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: x < z < y
There are several values of x, y and z that satisfy statement 2. Here are two:
Case a: x = 0, y = 3 and z = 1. In this case, 2z = 2(1) = 2 and x + y = 0 + 3 = 3. So, the answer to the REPHRASED target question is YES, 2z IS less than x+y
Case b: x = 0, y = 3 and z = 2. In this case, 2z = 2(2) = 4 and x + y = 0 + 3 = 3. So, the answer to the REPHRASED target question is NO, 2z is NOT less than x+y
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Hi All,

We're asked if the average (arithmetic mean) of the numbers X, Y, and Z is GREATER than Z. This is a YES/NO question and can be approached with a mix of Arithmetic and TESTing VALUES. To start, we can 'rewrite' the question a bit:

Is (X+Y+Z)/3 > Z?
Is (X+Y+Z) > 3Z?
Is (X+Y) > 2Z?

By comparison, this is an easier question to answer than what we were initially given.

(1) Z - X < Y - Z

With Fact 1, we can rewrite the inequality as:
2Z < X + Y
This Fact tells us that (X+Y) IS greater than 2Z, so the answer to the question is clearly YES.
Fact 1 is SUFFICIENT

(2) X < Z < Y

With this inequality, we can TEST VALUES and track the results.
IF....
X=1, Z=2, Y=3, then (1+3) is NOT greater than (2)(2), so the answer to the question is NO.
X=1, Z=2, Y=4, then (1+4) IS greater than (2)(2), so the answer to the question is YES.
Fact 2 is INSUFFICIENT

Final Answer: A

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