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If the average (arithmetic mean) of five distinct positive integers is

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Bunuel
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If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   03 Jul 2017, 03:40
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If the average (arithmetic mean) of five distinct positive integers is 10, what is the least possible value of the greatest of the five numbers?

(A) 11
(B) 12
(C) 24
(D) 40
(E) 46
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Abhishek009
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   03 Jul 2017, 09:01
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Bunuel wrote:
If the average (arithmetic mean) of five distinct positive integers is 10, what is the least possible value of the greatest of the five numbers?

(A) 11
(B) 12
(C) 24
(D) 40
(E) 46

\(a + ( a + 1 ) + ( a + 2 ) + ( a + 3 ) + (a + 4) = 50\) ( As the numbers are distinct )

Or, \(5a + 10 = 50\)

Or, \(5a = 40\)

Or, \(a = 8\)

So, The least value of the greatest of the five numbers is 8 + 4 = 12, answer , must be (B) 12
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   15 Nov 2017, 23:31
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shamsm wrote:
Hi,

I don't understand how to get the "least" possible value of the greatest. Could someone explain it please.

Thank you!


If the average (arithmetic mean) of five distinct positive integers is 10, what is the least possible value of the greatest of the five numbers?

(A) 11
(B) 12
(C) 24
(D) 40
(E) 46

Given: \(0<a < b < c< d< e\) and \(a + b + c + d + e = 10*5\).

We want to minimise e.

General rule for such kind of problems, when the sum is fixed:
to maximize one quantity, minimize the others;
to minimize one quantity, maximize the others.

So, here we should maximize a, b, c, and d. Since the integers are distinct, then the max value of d is e - 1, the max value of c is e - 2, the max value of b is e - 3 and the max value of a is e - 4.

Thus, \((e - 4) + (e - 3) + (e - 2) + (e - 1) + e = 50\);
e = 12.

Answer: B.


14. Min/Max Problems


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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   03 Jul 2017, 05:15
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Since the average of 5 numbers is 10, the total must be 50.
The numbers, must be consecutive for the greatest number to be least (8,9,10,11,12)

The least possible value of the greatest number(as it has been given that the numbers are distinct) has to be 12(Option B)
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   03 Jul 2017, 06:15
Ans :A
9,10,10,10,11

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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   03 Jul 2017, 06:18
Anannt wrote:
Ans :A
9,10,10,10,11

Sent from my Mi 4i using GMAT Club Forum mobile app


It cannot be 9,10,10,10,11 because it has been given that the integers are distinct.
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   15 Nov 2017, 21:51
Hi,

I don't understand how to get the "least" possible value of the greatest. Could someone explain it please.

Thank you!
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   30 Mar 2018, 00:04
Bunuel wrote:
shamsm wrote:
Hi,

I don't understand how to get the "least" possible value of the greatest. Could someone explain it please.

Thank you!


If the average (arithmetic mean) of five distinct positive integers is 10, what is the least possible value of the greatest of the five numbers?

(A) 11
(B) 12
(C) 24
(D) 40
(E) 46

Given: \(0<a < b < c< d< e\) and \(a + b + c + d + e = 10*5\).

We want to minimise e.

General rule for such kind of problems, when the sum is fixed:
to maximize one quantity, minimize the others;
to minimize one quantity, maximize the others.

So, here we should maximize a, b, c, and d. Since the integers are distinct, then the max value of d is e - 1, the max value of c is e - 2, the max value of b is e - 3 and the max value of a is e - 4.

Thus, \((e - 4) + (e - 3) + (e - 2) + (e - 1) + e = 50\);
e = 12.

Answer: B.


14. Min/Max Problems


Why 0<a<b<c<d<e?

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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   02 Apr 2018, 06:15
I don't understand the word "distinct". In my opinion, this word does not mean "consecutive". So, why do we should calculate

Thus, (e−4)+(e−3)+(e−2)+(e−1)+e=50. ?
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   04 Apr 2018, 07:56
AQHER wrote:
I don't understand the word "distinct".


The word distinct means that the numbers must not be same,

(Say) All the numbers are 2

Or Say, the sequence be 2, 2 , 1, 3 , 2

Thus distinct means that the numbers must not be equal.

AQHER wrote:
In my opinion, this word does not mean "consecutive". So, why do we should calculate

Thus, (e−4)+(e−3)+(e−2)+(e−1)+e=50. ?


Since the question stem asks " least possible value of the greatest of the five numbers " , we have assumed that the numbers are consecutive..

Hope this helps !!
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If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   Updated on: 04 Apr 2018, 10:16
Bunuel wrote:
If the average (arithmetic mean) of five distinct positive integers is 10, what is the least possible value of the greatest of the five numbers?

(A) 11
(B) 12
(C) 24
(D) 40
(E) 46



because sum (50) is multiple of 5, and integers are distinct, assume consecutive integers
checking options,
A: 4x+6=(50-11) no, x not integer
B: 4x+6=(50-12) yes
x=8
8,9,10,11,12
B

Originally posted by gracie on 04 Apr 2018, 09:58.
Last edited by gracie on 04 Apr 2018, 10:16, edited 3 times in total.
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   04 Apr 2018, 10:09
pushpitkc wrote:
Since the average of 5 numbers is 10, the total must be 50.
The numbers, must be consecutive for the greatest number to be least (8,9,10,11,12)

The least possible value of the greatest number(as it has been given that the numbers are distinct) has to be 12(Option B)


Why does the set have to be consecutive numbers? I understand everything up until the necessity to have a consecutive set since the question read distinct numbers.

Thank you in advance.
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   24 Jun 2022, 22:02
Bunuel wrote:
If the average (arithmetic mean) of five distinct positive integers is 10, what is the least possible value of the greatest of the five numbers?

(A) 11
(B) 12
(C) 24
(D) 40
(E) 46


We can use summation formula; sum= n/2 (2a+(n-1)d). 5/2(2a+ 4 x d) = 50; a+2d= 10. a has to start with 2, as all the number are integers(distinct). If a=1, d will be in fraction.

Possible combination: a=2, d=4; a=4, d=3; a=6, d=2; a=8, d=1.

possible number for the above sequence: (2,6,10,14,18), (4,7,10,13,16), (6, 8,10,12,14) & (8,9,10,11,12). So the least highest is 12.
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   24 Jun 2022, 22:33
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Bunuel wrote:
If the average (arithmetic mean) of five distinct positive integers is 10, what is the least possible value of the greatest of the five numbers?

(A) 11
(B) 12
(C) 24
(D) 40
(E) 46


We can also test answer choices on questions like this.

A) If the greatest is 11, how are we going to get to an average of 10? The highest we could have is 11, 10, 9, 8, 7, which doesn't work. We are close, but (spoiler alert!) need something just a little bigger. Eliminate.
B) Adjusting what we just did for A, now we have 12, 11, 10, 9, 8. Yay!!

Answer choice B.
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Re: If the average (arithmetic mean) of five distinct positive integers is [#permalink] New post   27 Jun 2023, 07:15
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