If (x + y)/ z

Question

If (x + y)/ z < 0 and z not equal to 0, then is the average of x, y, and z greater than zero?

(1) z = -5 and the range of x, y, and z is 6.

(2) x + y = 2 and the range of x, y, and z is 6.

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kungfury42 · Answer

(x+y)/z < 0 means one of (x+y) or z is negative, and the other is positive. We have to answer whether their average (or sum) is positive or is the value of the positive quantity greater than the absolute value of the negative quantity.

(1) z=(-5), Range=6

This means, x+y will be positive
Which means at least 1 of x or y must be positive. The other could be positive or a negative but of a smaller absolute magnitude.

This can be feasible only if (-5) is one extreme end of the range, meaning (-5)+6=1 is the other extreme end of the range. If 1 is one of x or y, the other one will lie between (-1, 1].

Even if the other quantity is 1, the sum x+y will be at max 2 and still smaller than the absolute value of z. Which means the sum x+y+z overall can never be positive. Which means the sum and the average is always negative. Hence, sufficient.

(2) x+y=2, Range=6

Since x+y=2, then z must be negative.

Case1: x, y are both +ve, z -ve
x=y=1 and z=(-5)
x+y+z=(-3) implies average<0

Case2: x>0, y<0, z -ve
x=4, y=(-2), z=(-1)
x+y+z=1 implies average>0

Statement 2 is insufficient.

Hence, only statement 1 is sufficient in answering the question, option A.

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Sonia2023 · Answer

chetan2u  - can you please explain this question?

chetan2u · Answer

We are looking at:  \frac{x+y+z}{3}>0……..x+y+z>0 
So, the question is whether sum of the three number is positive when x+y and z are of opposite sign.

(1) z = -5 and the range of x, y, and z is 6.
Since range is 6, the maximum value of x or y is -5+6 or 1.
Even when we take the largest possible alive of x and y,  x+y+z=1+1-5=-3 .
Thus, the sum is always negative and answe is NO. 
Sufficient

(2) x + y = 2 and the range of x, y, and z is 6.
Let us take x and y as extreme points so that z could be any value between them. 
So  x=y+6 or (y+6)+y=2…….y=-2…..x=-2+6=4 
Now, z can be anything from -2 to 4.
When  z=4……x+y+z=4-2+4=6,  and answer is YES
When  z=-2……x+y+z=4-2-2=0,  and answer is NO
(You could take 1+1-5 too and you will get a NO.)
Insufficient

A

If (x + y)/ z < 0 and z not equal to 0, then is the average of x, y, a

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