Set S contains seven distinct integers. The median of set S

Question

Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

A. m
B. 10m/7
C. 10m/7 – 9/7
D. 5m/7 + 3/7
E. 5m

A. m 
B. 10m/7 
C. 10m/7 – 9/7 
D. 5m/7 + 3/7 
E. 5m

Bunuel · Accepted Answer

If a set has odd number of terms the median of a set is the middle number when arranged in ascending or descending order;
If a set has even number of terms the median of a set is the average of the two middle terms when arranged in ascending or descending order.

So median of S, which contains seven terms is 4th term when arranged in ascending order;:  median=4th \ term=m .

Now, to maximize the mean we should maximize the terms. As numbers in S are  distinct integers  and the highest number in S could be equal to  2m , then maximum values of the terms would be:  m-3 ,  m-2 ,  m-1 ,  median=m ,  2m-2 ,  2m-1 ,  2m .

Mean=\frac{(m-3)+(m-2)+(m-1)+m+(2m-2)+(2m-1)+2m}{7}=\frac{10m-9}{7} .

Answer: C.

saxenashobhit · Answer

I had missed reading distinct and made the set as m,m,m,m,2m,2m,2m and came with option B. Phuf...why they keep such answer choices

LJ · Answer

Seven distinct integers, the highest value is equal to twice the median. To achieve the highest possible mean you need the following seven integers:

Set S =  {2m, 2m - 1, 2m - 2, m, m - 1, m - 2, m - 3}

10m-\frac{9}{7} =  \frac{10m}{7} - \frac{9}{7}

Answer is C

gmatprav · Answer

To maximize the average we need to maximize all the  distinct integers  in the set.

the numbers then will be

m-3, m-2, m-1,m,2m-2,2m-1, 2m average is sum of these divided by 7.

\frac{(10m-9)}{7} = \frac{10m}{7}- \frac{9}{7}

C.

Lucky2783 · Answer

Mean=\frac{(m-3)+(m-2)+(m-1)+(m)+(2m-2)+(2m-1)+(2m)}{7} = \frac{10m-9}{7} 
Answer :C

thefibonacci · Answer

just assume m = 7 (median)

now since each term is distinct and max is 2m, to achieve max sum ... terms should be 4,5,6,7,12,13,14 
mean = 61/7

by options, C is satisfied. so C.

shasadou · Answer

looks like picking numbers won't help and one should solve with algebra

tuandoan92 · Answer

Hi, i got the right answer but i am a bit concern. The question says all all values in set S are equal to or less than 2m. So is that the maximum value should be 2m-1 because it obviouly says the value is less then 2m. Plz help me. I put 2m-1 for the first time and it is not in the showed answer, so i have to take 2m as a maximum value. Thanks

pakasaip · Answer

one word that change the whole game . . "distinct"

ScottTargetTestPrep · Answer

Since there are 7 distinct integers, there are 3 integers below the median and 3 above. Furthermore, since we want the largest possible average of these integers, we want the integers to be as large as possible. Thus we can let the largest integer be 2m, the second largest (2m - 1), and the third largest (2m - 2). The fourth largest is the median, so it must be m. The fifth, sixth, and seventh (or the smallest) integers will be (m - 1), (m - 2), and (m - 3), respectively. Thus, the largest possible average is:

/7

(10m - 9)/7

10m/7 - 9/7

Answer: C

BrentGMATPrepNow · Answer

One approach is to TEST a value of m.

Let's say m =  5 .
So, when we arrange all 7 values in ASCENDING order, 5 is the MEDIAN: _ _ _  5  _ _ _
Since all values in set S are equal to or less than 2m, the BIGGEST value is 10. 
So, we get: _ _ _  5  _ _ 10
At this point, we are tying to MAXIMIZE the other values AND make sure all are DISTINCT. 
So, we get: 2, 3, 4,  5 , 8, 9, 10
The average = (2 + 3 + 4 + 5 + 8 + 9 + 10)/7 =  41/7

Now plug m =  5  into the answer choices to see which one yields an average of  41/7

A)  5  NOPE
B) 10m/7. So, we get: 10( 5 )/7 =  50/7  NOPE
C) 10m/7 – 9/7. So, we get: 10( 5 )/7 - 9/7 =  41/7  BINGO!!
D) 5m/7 + 3/7. So, we get: 5( 5 )/7 + 3/7 =  28/7  NOPE
E) 5m. So, we get: 5( 5 ) =  25  NOPE

Answer:  C

Cheers,
Brent

Alecita · Answer

I used Picking Numbers to solve this question.

M= 4 Median

2M= 8 All values are equal or less than 8

This way, the 7 different integers in Set S are: 1, 2, 3, 4, 6, 7, 8

AVG = \(\frac{sum of terms}{# of terms}\) = \(\frac{1+2+3+4+6+7+8}{7}\) = \(\frac{31}{7}\)

When we plug M=4 in the answer choices, we have to find the match AVG = \(\frac{31}{7}\)

A) 4 not a match

B) \(\frac{40}{7}\) not a match

C) \(\frac{40}{7}\) - \(\frac{9}{7}\) = \(\frac{31}{7}\) that's a match

D) \(\frac{20}{7}\) + \(\frac{3}{7}\) = \(\frac{23}{7}\) not a match

E) 20 not a match

Hence, answer C is the correct choice.

Hope it helps!

Thanks! Alecita

Shreenidhi24 · Answer

@Bunel,
To maximise the average, the difference between each number should be 1 starting from the highest number, which is 2m. So, I calculated 2m, 2m-1, 2m-2, 2m-3 ( Which is m), 2m-4, 2m-5, 2m-6 and got a different answer. Can you help me understand where I went wrong?

Archit3110 · Answer

sequence of the terms would be with median i.e 4th term as m
m-3,m-2,m-1,m,2m-2,2m-1,2m
2m is highest term
sum ; 10m-9
avg ; 10m/7-9/7
option C

MathRevolution · Answer

Total numbers: 7

Median: 4th number: m

The highest number equal to 2m. Therefore the smallest number will be m - 3

Average:  \frac{  }{ 7}

=> Average:  \frac{  }{7}

=> Average:  \frac{10m }{ 7} - \frac{9}{ 7}

Answer C

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Set S contains seven distinct integers. The median of set S

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Set S contains seven distinct integers. The median of set S

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