Is the average of n consecutive integers equal to 1 ?

Question

(1) n is even 
(2) if S is the sum of the n consecutive integers, then 0 < S < n

Correct answer D

Hello Fluke and Everyone in this forum - can you please explain the conceptual way of to solve this and improve my understanding ?

My approach to solve this:

Statement 1: Sufficient, because the average of the even number of consecutive integers can never be an integer.

Statement 2:  I understand what the statement means, but I am struggling to understand the concept.

My understanding is

Average = Sum/Number of terms
Therefore, Sum = Average * Number of Terms
Therefore S=Average/n. But what can we conclude from this? The book says that this is sufficient as well.

So the answer is D. But I don't understand the Statement 2.

stonecold · Accepted Answer

Great Question.
Here is what i did in this one ->

We are asked if the mean of n consecutive is an integer or not.

N consecutive integer => Evenly spaced set =>Mean = Median =Average of the first and the last term

Mean for N consecutive integers can take only 2 forms =>
x => For N = odd
x.5=> for N=even

for some integer x

Statement 1-->
N is even

So mean will never be an integer(Because median will never be an integer)
Hence mean can never be "one".
Hence Sufficient

Statement 2-->
Sum=> S
And 0<S<N

\(Mean = \frac{Sum}{#}\)

Mean = \(\frac{S}{N}\)

As \(0<S<N\)=>

\(\frac{S}{N}\) will be a decimal between (0,1)

Hence mean will never be one.
Hence Sufficient

Hence D

fluke · Answer

2) To get an average=1; the Sum of elements must be equal to number of elements.

If S=n; \frac{S}{n}=1 ;

If S is NOT equal to n, the average can not be 1.

Sum=100; n=100; S/n=100/100=1
Sum=1; n=1; S/n=1/1=1
Sum=10; n=9; S/n=10/9=1.##(NOT EQUAL TO 1)

Sufficient.

Bunuel · Answer

Is the average of n consecutive integers equal to 1 ?

(1) n is even
(2) if S is the sum of the n consecutive integers, then 0 < S < n

Show SpoilerMy approach

Correct answer D

Hello Fluke and Everyone in this forum - can you please explain the conceptual way of to solve this and improve my understanding ?

My approach to solve this:

Statement 1: Sufficient, because the average of the even number of consecutive integers can never be an integer.

Statement 2: I understand what the statement means, but I am struggling to understand the concept.

My understanding is

Average = Sum/Number of terms
Therefore, Sum = Average * Number of Terms
Therefore S=Average/n. But what can we conclude from this? The book says that this is sufficient as well.

So the answer is D. But I don't understand the Statement 2.

Show Answer

Official Answer

D

woohoo921 · Answer

KarishmaB I hope that you are having a wonderful week

I was confused as to how you can just divide the whole inequality by n for statement 2. I have only seen manipulation of inequalities when you divide the middle term in the inequality (e.g., in which the inequality would be divided by s in this case --> 0/s < 1 < n/s). To clarify, you can divide by any term in the inequality as long as you consistently divide each variable or expression in the inequality by that term (except I guess in this example you cannot divide the inequality by the first term because it is a 0). Many thanks

KarishmaB · Answer

You can divide an inequality by any positive number without impacting the inequality. 
You might have seen division by middle term (provided we know the term is positive) because in that question it would have helped simplify. Otherwise, you can divide by any positive number.

a < b
a/2 < b/2 (valid)

a < b < c
a/10 < b/10 < c/10 (valid)

a < b < c
1 < b/a < c/a (valid if a is known to be a positive number)

etc.

woohoo921 · Answer

Bunuel I thought that we can only work with manipulating the inner term (e.g., s) in this case per my question above for Karishmab In other words, if I have: 5 < x+2 < 9 Can you do the following for division? Dividing all terms by 5 --> 1 < (x+2)/5 < 9/5 Dividing all terms by x+2 --> x/(x+2) < 1 < 9/(x+2) Dividing all terms by 9 --> 5/9 < (x+2)/9 < 1 Can you do the following for subtraction? Again, if I have: 5 < x+2 < 9 Subtracting 9 from all terms --> -4 < x-7 < 0 Subtracting x+2 from all terms --> -x+3 < 0 < -x-2 and vice versa for multiplication? Many thanks

KarishmaB

MacroMX · Answer

I think Statement 1 is not sufficient: assume -1;0;1;2 here average is 1, 1;2;3;4 here average is 5. Isn't it insufficient

KarishmaB · Answer

We can add or subtract the same term on both sides of the inequality without changing the sign of the inequality. We can multiply/divide the same term on both sides without impacting the inequality sign provided the term is positive. 
Here, since (x + 2) lies between 5 and 9, we know it must be positive. So we can multiply or divide the inequality by (x + 2) and the inequality sign remains the same.

KarishmaB · Answer

When we have even number of consecutive integers, average will be the average of the two middle integers so it will not be an integer.

Avg of -1, 0, 1, 2 is the average of 0 and 1 which is 0.5.
Avg of 1, 2, 3, 4 is the average of 2 and 3 which is 2.5.

woohoo921 · Answer

KarishmaB

I thought that you can't multiply nor divide inequalities together, or does that rule only apply to when you have two separate inequalities?

KarishmaB · Answer

We can multiply and divide inequalities as long as both sides are non negative in both inequalities (and there is no 0 in the denominator in case of division).
We can multiply when both inequalities have the same inequality sign and we can divide when they have opposite inequality signs.

A < B
C < D
-----
AC < BD - valid if A, B, C and D are all non negative

A < B
C > D
-----

A/C < B/D - valid if A and B are non negative and C and D are positive.

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.

Is the average of n consecutive integers equal to 1 ?

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 Data Sufficiency (DS) Questions

Is the average of n consecutive integers equal to 1 ?

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