A set consists of n positive integers arranged in decreasing order ...

Question

A set consists of n positive integers arranged in decreasing order of value. If the difference between any two adjacent numbers of the set is 3, then what is the ratio of arithmetic mean of the set to the median of the set?

A.

B.

C.

D.

E.

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A.  1
 B.  1.5
 C.  2
 D.  2.5
 E.  3

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Archit3110 · Answer

test with both odd & even set of n integers in set n
n= ( 12,9,6,3) and n= (12,9,6)

for n even integers
mean = 30/4 = 7.5 and median 6+9/2 = 7.5
ratio = 7.5/7.5 = 1

for n odd integers 
mean = 27/3 = 9 and median 9
ratio = 9/9 = 1
IMO A ; 1

jfranciscocuencag · Answer

My approach was that in an arithmetic progression median = mean.

Hence A.

Or, Am I wrong?

Kind regards!

EgmatQuantExpert · Answer

Solution 
  Given:  
 • A set of n positive integers arranged in decreasing order of value
• Difference between any two consecutive numbers of the sequence = 3

To find:  
 • The ratio of arithmetic mean of the set to the median value of the set

Approach and Working:   
 • We know that the median of the set depends upon the number of terms in the set.

Case 1 : n is odd
The terms of the set can be written as {a, a – 3, a – 3*2, a – 3*3, …. , a – 3(n-1)}
 • Median = the middle term =  (n + \frac{1}{2})^{th} term = a – 3  = a – \frac{3(n – 1)}{2} 
• Mean = (first term + last term)/2 =  a + a – \frac{3(n – 1)}{2} = a – \frac{3(n – 1)}{2} 
• Thus, median = mean

Case 2 : n is even
The terms of the set can be written as {a, a – 3, a – 3*2, a – 3*3, …. , a – 3(n-1)}
 • Median = average of the middle two terms =  \frac{1}{2} *   = \frac{1}{2} *   + a – 3(\frac{n}{2} + 1 – 1)] = \frac{(2a - 3n + 3)}{2} = a – \frac{3(n – 1)}{2} 
• Mean = (first term + last term)/2 =  a + a – \frac{3(n – 1)}{2} = a – \frac{3(n – 1)}{2} 
• Thus, median = mean

Therefore, for any value of n, the ration of median to mean of the set is 1

Hence, the correct answer is option A.

Answer: A

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Prerana94 · Answer

That is correct. They both are same in A.P

CrackverbalGMAT · Answer

CONCEPT: - Mean and Median - Arithmetic Mean = Sum of observations/Number of Observations - Median = Middle value in the set of observations. - Arithmetic sequence of numbers have the property that the Mean and Median of the set are the same number. The mean of two numbers on the number line is the midpoint. For example , the mean of 1 & 7 is 4 because the distance from 1 to 4 is same as the distance between 4 and 7. In an Arithmetic sequence the midpoint between the first and last term is exactly the median. SOLUTION: In this question, the sequence of positive integers with a difference of 3 is an Arithmetic sequence. Hence the mean is same as median. Thus, the ratio of Mean to Median is 1:1. (A) Hope this helps.

Keep studying.

Devmitra Sen

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