If p, r, and s are consecutive integers in ascending

The average of an equally spaced set is equal to the median of that set, so here, if p, r, s are consecutive integers in increasing order, their average is simply r, so r = x and the question is just asking if we can find r. Notice also that the sum of the three integers is therefore 3r (because by the definition of an average, the sum equals the average times the number of terms).

So Statement 1 just tells us that 2r = 3r, or that r = 0, and is sufficient.

Statement 2 tells us that 3r = 0, or that r = 0, and is also sufficient.

AbeNeedsAnswers wrote:If p, r, and s are consecutive integers in ascending order and x is the average (arithmetic mean) of the three integers, what is the value of x ?

(1) Twice x is equal to the sum of p, r, and s.
(2) The sum of p, r, and s is zero.

D

Given: p, r, and s are consecutive integers in ascending order and x is the average (arithmetic mean) of the three integers
Since p, r and s are EQUALLY spaced, the mean of the 3 numbers = the median of the 3 numbers.
Since p < r < s, we know that r = the mean = the median.

In other words, r = x (since we're told x is the mean)
So, p = x - 1
And s = x + 1 (since p, r, and s are consecutive integers)

Target question: What is the value of x?

Statement 1: Twice x is equal to the sum of p, r, and s.
We can write: 2x = p + r + s
Replace p with x-1, replace r with x, and replace s with x+1 to get: 2x = (x-1) + x + (x+1)
Simplify: 2x =3x
Solve: x = 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The sum of p, r, and s is zero.
We can write: p + r + s = 0
Replace p with x-1, replace r with x, and replace s with x+1 to get: (x-1) + x + (x+1) = 0
Simplify: 3x =0
Solve: x = 0
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

Hi All,

We're told that P, R and S are CONSECUTIVE integers in ASCENDING order and X is the AVERAGE (arithmetic mean) of the three integers. We're asked for the value of X.

To start, it's worth noting that since we're dealing with 3 CONSECUTIVE integers, the sum will always be '3 times' the value of the 'middle' numbers, so the average of those 3 numbers will be the value of the "middle" number - in this case, "R" will be the average. This means that X = R, so if we find the value of X or R, then we have answered the question. We can also set up the 'Average Formula': (P + R + S)/3 = X. This can be rewritten into any of the following:

P + R + S = 3X
P + R + S = 3R
P + S = 2R

This question can be approached in a couple of different ways; here's a way to use a mix of Algebra and Number Properties.

(1) Twice X is equal to the sum of P, R and S.

With the information in Fact 1, we can create the following equation:
2X = P + R + S

Comparing this equation to the initial equations we wrote, something interesting should stand out: How can (P + R + S) be equal to 3X AND 2X. Those appear to be 2 different results... and there's only one situation in which they are NOT.... when X = 0. Thus, the 3 values would be -1, 0 and 1 and the average is 0.
Fact 1 is SUFFICIENT

(2) The sum of P, R and S is zero.

Fact 2 gives us the SUM of the three terms, so we can easily calculate the average... X = (0)/3 = 0.
Fact 2 is SUFFICIENT

Final Answer: D

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