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Math Expert V
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If the average (arithmetic mean) of five distinct positive integers is  [#permalink]

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Question Stats: 61% (01:35) correct 39% (01:27) wrong based on 231 sessions

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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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Posts: 58344
Re: If the average (arithmetic mean) of five distinct positive integers is  [#permalink]

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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.

14. Min/Max Problems

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Re: If the average (arithmetic mean) of five distinct positive integers is  [#permalink]

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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]

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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]

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Ans :A
9,10,10,10,11

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Re: If the average (arithmetic mean) of five distinct positive integers is  [#permalink]

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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]

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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]

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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.

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]

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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]

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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]

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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

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]

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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]

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_________________ Re: If the average (arithmetic mean) of five distinct positive integers is   [#permalink] 22 Apr 2019, 01:07
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