If the average (arithmetic mean) of a list of four integers remains

Question

If the average (arithmetic mean) of a list of four integers remains unchanged when all the integers are multiplied by any constant, which of the following statements must be true?

I. I. The average (arithmetic mean) of the list is 0.

II. The sum of the largest and smallest members of the list is 0.

III. The list contains both positive and negative integers.

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Bunuel · Accepted Answer

Official Solution:

If the average (arithmetic mean) of a list of four integers remains unchanged when all the integers are multiplied by  any  constant, which of the following statements must be true?
 
I. I. The average (arithmetic mean) of the list is 0.
 
II. The sum of the largest and smallest members of the list is 0.
 
III. The list contains both positive and negative integers.

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Since the mean remains unchanged when all the integers of the list are multiplied by any constant, it must also remain unchanged when all the integers are multiplied by 0. If we multiply all integers by 0, we get a list containing only 0's: {0, 0, 0, 0}, which has a mean of 0. Therefore, the mean of the original list must also be 0. 
 
I. I. The average (arithmetic mean) of the list is 0. As discussed earlier, this statement must be true.
 
II. The sum of the largest and smallest members of the list is 0. This statement is not necessarily true. We can have a list with a mean of 0, where the sum of the largest and smallest members is not 0. For example, consider the list {-3, 0, 1, 2}.
 
III. The list contains both positive and negative integers. This statement is not necessarily true, as the list can consist of only zeros.

Answer: A

Archit143 · Answer

I have a doubt in this question
If we consider a set {1,2,3,4}, when multiplied with a constant we take that constant as 1 and multiply resulting set will contain the same numbers as the original. so how can the answer be I only.....

KarishmaB · Answer

The question tells you that 'The mean of four integers will not change if all the integers are multiplied by  any  constant.'

This means that when you multiply all the four integers by any constant e.g. 1 or 2 or 5 or 100 or -20 etc, you will always get the same mean.
In case of (1, 2, 3, 4}, the mean stays the same only when you multiple each number by 1. When you multiply each number by some other number e.g. 2, the mean changes. So {1, 2, 3, 4} doesn't satisfy our condition.

If the mean is 0, all the numbers will add up to 0.
a+b+c+d = 0
When you multiply this sum by any constant, the sum will remain 0 and hence the new mean will remain 0.

EMPOWERgmatRichC · Answer

Hi All,

In Roman Numeral questions, it's often easiest to DISPROVE the Roman Numerals, if possible (so that you can eliminate answer choices).

Here, we're told that the MEAN of 4 integers will NOT change if each integer is multiplied by the same (re: ANY) constant. We're asked which of the Roman Numerals is ALWAYS TRUE. We can TEST VALUES to eliminate options....

II. The sum of the largest and smallest is 0

IF...the set is {-3, 0, 1, 2} then the mean would stay the same, but the sum of the largest and smallest is NOT 0. 
Roman Numeral II is NOT true. 
Eliminate Answers B, D and E

III. The set contains BOTH positive AND negative numbers

IF....the set is {0, 0, 0, 0} then the mean would stay the same and the set does NOT contain both positive and negative numbers. 
Roman Numeral III is NOT true. 
Eliminate Answer C.

There's only one answer left....

Final Answer:  A

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If the average (arithmetic mean) of a list of four integers remains

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If the average (arithmetic mean) of a list of four integers remains

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards