If p, r, and s are consecutive integers in ascending order and x is th

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If x is the average of three consecutive integers p, r and s, then it’s the value of r, the middle integer.

Statement One Alone:

Twice x is equal to the sum of p, r, and s.

We can create the equation:

2x = p + r + s

2x = (r - 1) + r + (r + 1)

2x = 3r

Since we have concluded x = r in the stem, we have:

2x = 3x

x = 0

Statement one alone is sufficient to answer the question.

Statement Two Alone:

The sum of p, r, and s is zero.

Since x is the average of p, r and s, we have:

x = (p + r + s )/3 = 0/3 = 0

Statement two alone is sufficient to answer the question.

Answer: D

Solution

Steps 1 & 2: Understand Question and Draw Inferences

In this question, we are given

• The numbers p, r, and s are consecutive integers in ascending order.
• x is the average (arithmetic mean) of the three integers.

We need to determine

• The value of x.

As x is the average of p, r, and s, we can write x = \(\frac{(p+r+s)}{3}\)

Therefore, to find the value of x, we need to know either the individual values of p, r, and s, or their sum.
With this understanding, let us now analyse the individual statements.

Step 3: Analyse Statement 1

As per the information given in statement 1, twice x is equal to the sum of p, r, and s.

• 2x = p + r + s

We also know that 3x = p + r + s
Therefore, we can write 2x = 3x

• Or, x = 0

As we can determine the value of x, statement 1 is sufficient to answer the question.

Step 4: Analyse Statement 2

As per the information given in statement 2, the sum of p, r, and s is zero.

• p + r + s = 0

As x = \(\frac{(p+r+s)}{3}\), we can say

• x = \(\frac{0}{3}\) = 0

As we can determine the value of x, statement 2 is sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)
Since we can determine the answer from either of the statements individually, this step is not required.
Hence, the correct answer choice is option D.

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:

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

Dear All,

Thank you for sharing the question and solution.

My only question is that it is classified as an easy question in OG 2020 and in GMAT Club it is classified as an 700 level question. Any reason for discrepancy in classification between OG and GMAT club.

Hello,
What is my mistake if I do the following to rewrite the consecutive numbers:
p= x
r= x+1
s= x+2
so, the average (x+x+1+x+2)/3=x
3x+3 = 3x
and getting nothing.
what's wrong with me?

Hi Baten80,

When you have 3 CONSECUTIVE INTEGERS, the average of those 3 numbers will be the 'middle number.' The prompt specifically refers to the average as 'X', so you would have to write the numbers as:

X - 1
X
X + 1

In your example, you made the SMALLEST number equal to X, but that doesn't 'fit' the rest of the information that you're given (and why the last line of your work doesn't make sense).

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

Thanks Rich! Do you mind going a little bit more detail on this concept "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"please?

I'm clear on mean = median but I haven't come across this sort of shortcut before.

Cheers

Hi adkor95,

You can actually prove that this is a pattern by choosing any 3 consecutive integers (and you can even use negative integers and/or 0, if you want). Add them up and the result will ALWAYS be 3 times the 'middle' number.

Algebraically, if we called the three numbers X, (X+1) and (X+2), then the sum would be X + X+1 + X+2 = 3X + 3. Here, the middle term is (X+1) and that sum (3X + 3) is 3 times that number.

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

I answered this question correctly but I took 3.03 minutes instead of less than 1 minute. I want to mention the mistakes besides my correct analogeis.

Given that p, r, and s consecutive numbers in ascending order. We know that for an arranged consecutive numbers the middle number is the average. So, \(r\) is the average, which is the \(x\) in this question.

So, x will be in the middle position, the left one will be \(x-1\) and the right one will be \(x+1\), now the arrangement is \(x-1, x, \ and / x+1\). Don't mistake considering \(p, r, \ and \ s \ as \ x, x+1,\) and \(x+2\) as the middle number is \(x+1\) which is not equal to \(x\). I did this mistake.

Thus, the bottom line is:

\(r=x\)

\(p=x-1\) (as it's the previous number of x)

\(s= x+1\) (as it's the post number of x)

The numbers are: \((x-1), \ x, \ (x+1)\)

(1) \(x-1+x+x+1=2x\)

\(3x=2x\)

\(x=0\); Sufficient.

Twice x is equal to the sum of p, r, and s.

(2) \(x-1+x+x+1=0\)

\(3x=0\)

\(x=0;\) Sufficient.

The answer is \(D\)

When I was stuck to solve the DS. I took a different approach. I looked at the conditions and saw that the second condition said that the total of the integers is 0. Look here, if the total is \(0\) then the average is \(0\), which is the middle number as per the given condition. We already got \(x=0\). For my further understanding, the previous number is \(0-1=-1\) and the post number is \(0+1=1,\) the numbers are \(-1,0,1.\)
Taking this example also satisfies the first condition. \(-1+0+1=0, \ then \ x=0. \)Sufficient.

The answer is \(D \)

I answered this question correctly but I took 3.03 minutes instead of less than 1 minute. I want to mention the mistakes besides my correct analogeis.

Given that p, r, and s consecutive numbers in ascending order. We know that for an arranged consecutive numbers the middle number is the average. So, \(r\) is the average, which is the \(x\) in this question.

So, x will be in the middle position, the left one will be \(x-1\) and the right one will be \(x+1\), now the arrangement is \(x-1, x, \ and / x+1\). Don't mistake considering \(p, r, \ and \ s \ as \ x, x+1,\) and \(x+2\) as the middle number is \(x+1\) which is not equal to \(x\). I did this mistake.

Thus, the bottom line is:

\(r=x\)

\(p=x-1\) (as it's the previous number of x)

\(s= x+1\) (as it's the post number of x)

The numbers are: \((x-1), \ x, \ (x+1)\)

(1) \(x-1+x+x+1=2x\)

\(3x=2x\)

\(x=0\); Sufficient.

Twice x is equal to the sum of p, r, and s.

(2) \(x-1+x+x+1=0\)

\(3x=0\)

\(x=0;\) Sufficient.

The answer is \(D\)

When I was stuck to solve the DS. I took a different approach. I looked at the conditions and saw that the second condition said that the total of the integers is 0. Look here, if the total is \(0\) then the average is \(0\), which is the middle number as per the given condition. We already got \(x=0\). For my further understanding, the previous number is \(0-1=-1\) and the post number is \(0+1=1,\) the numbers are \(-1,0,1.\)
Taking this example also satisfies the first condition. \(-1+0+1=0, \ then \ x=0. \)Sufficient.

The answer is \(D \)

surprisingly, OG graded this question as an easy level.

Answer: Option D

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For statement 1:
The 3 numbers can be 0,1,2 which will give us the value of x as 1.5
2x = sum of integers ..which is also correct
or x= 0.
why was statement 1 sufficient?

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