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e-GMAT Representative D
Joined: 04 Jan 2015
Posts: 2888
A set consists of n positive integers arranged in decreasing order ...  [#permalink]

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Difficulty:   15% (low)

Question Stats: 84% (01:32) correct 16% (01:49) wrong based on 71 sessions

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

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Concentration: Sustainability, Marketing
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Re: A set consists of n positive integers arranged in decreasing order ...  [#permalink]

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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

EgmatQuantExpert wrote:
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. 1
B. 1.5
C. 2
D. 2.5
E. 3

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Senior Manager  S
Joined: 12 Sep 2017
Posts: 267
Re: A set consists of n positive integers arranged in decreasing order ...  [#permalink]

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

Hence A.

Or, Am I wrong?

Kind regards!
e-GMAT Representative D
Joined: 04 Jan 2015
Posts: 2888
Re: A set consists of n positive integers arranged in decreasing order ...  [#permalink]

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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 = $$(n + \frac{1}{2})^{th} term = a – 3[\frac{(n + 1)}{2} – 1] = a – \frac{3(n – 1)}{2}$$
• 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 = $$\frac{1}{2} * [(\frac{n}{2})^{th} term + (\frac{n}{2} + 1)^{th} term] = \frac{1}{2} * [a – 3[(\frac{n}{2} – 1] + a – 3(\frac{n}{2} + 1 – 1)] = \frac{(2a - 3n + 3)}{2} = a – \frac{3(n – 1)}{2}$$
• 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.

_________________ Re: A set consists of n positive integers arranged in decreasing order ...   [#permalink] 11 Jan 2019, 07:10
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