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# If 6 positive integers have a sum less than 10, what is the value of

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Math Expert
Joined: 02 Sep 2009
Posts: 43894
If 6 positive integers have a sum less than 10, what is the value of [#permalink]

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09 Jan 2018, 22:38
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If 6 positive integers have a sum less than 10, what is the value of the greatest of the integers?

(1) The average (arithmetic mean) of the 6 integers is 3/2.
(2) The median of the integers is 1.
[Reveal] Spoiler: OA

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Re: If 6 positive integers have a sum less than 10, what is the value of [#permalink]

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09 Jan 2018, 22:53
Bunuel wrote:
If 6 positive integers have a sum less than 10, what is the value of the greatest of the integers?

(1) The average (arithmetic mean) of the 6 integers is 3/2.
(2) The median of the integers is 1.

Let the integers in ascending order be a, b, c, d, e, f. We are given that a+b+c+d+e+f < 10. WE have to find 'f'.

(1) Avg = 3/2. So sum = 3/2 * 6 = 9. This means
a+b+c+d+e+f = 9. So there is a possibility that a=b=c=d=e=1, and then f=4. Or we could have a=b=c=1, and d=e=f=2. Multiple possibilities for 'f'. So Insufficient.

(2) Median is 1. So the average of middle two integers (c &d) is 1. Which means c+d = 2. This is only possible when c=d=1. And since they are in ascending order, this also means that a=b=1 too (no integer can be 0 since they are all positive integers). So a=b=c=d=1. This means e+f < 6 (total sum has to be less than 10). But we could have e=1, f=4 Or we could have e=2, f=3. Multiple possibilities for 'f'. Insufficient.

Combining the statements, we get that a=b=c=d=1. And now e+f=5. But again if e=1, f=4 and if e=2, f=3. WE cant uniquely determine value of 'f'. Insufficient.

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Re: If 6 positive integers have a sum less than 10, what is the value of [#permalink]

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11 Jan 2018, 07:18
amanvermagmat wrote:
Bunuel wrote:
If 6 positive integers have a sum less than 10, what is the value of the greatest of the integers?

(1) The average (arithmetic mean) of the 6 integers is 3/2.
(2) The median of the integers is 1.

Let the integers in ascending order be a, b, c, d, e, f. We are given that a+b+c+d+e+f < 10. WE have to find 'f'.

(1) Avg = 3/2. So sum = 3/2 * 6 = 9. This means
a+b+c+d+e+f = 9. So there is a possibility that a=b=c=d=e=1, and then f=4. Or we could have a=b=c=1, and d=e=f=2. Multiple possibilities for 'f'. So Insufficient.

(2) Median is 1. So the average of middle two integers (c &d) is 1. Which means c+d = 2. This is only possible when c=d=1. And since they are in ascending order, this also means that a=b=1 too (no integer can be 0 since they are all positive integers). So a=b=c=d=1. This means e+f < 6 (total sum has to be less than 10). But we could have e=1, f=4 Or we could have e=2, f=3. Multiple possibilities for 'f'. Insufficient.

Combining the statements, we get that a=b=c=d=1. And now e+f=5. But again if e=1, f=4 and if e=2, f=3. WE cant uniquely determine value of 'f'. Insufficient.

Hi amanvermagmat, chetan2u

The question asks for the greatest value of the integer which can be only one value i.e. 4. Other possibilities are there but of all the possibilities greatest is 4. Can you please let me know if I am missing something.
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Joined: 02 Aug 2009
Posts: 5660
If 6 positive integers have a sum less than 10, what is the value of [#permalink]

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11 Jan 2018, 07:24
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Expert's post
rahul16singh28 wrote:
amanvermagmat wrote:
Bunuel wrote:
If 6 positive integers have a sum less than 10, what is the value of the greatest of the integers?

(1) The average (arithmetic mean) of the 6 integers is 3/2.
(2) The median of the integers is 1.

Let the integers in ascending order be a, b, c, d, e, f. We are given that a+b+c+d+e+f < 10. WE have to find 'f'.

(1) Avg = 3/2. So sum = 3/2 * 6 = 9. This means
a+b+c+d+e+f = 9. So there is a possibility that a=b=c=d=e=1, and then f=4. Or we could have a=b=c=1, and d=e=f=2. Multiple possibilities for 'f'. So Insufficient.

(2) Median is 1. So the average of middle two integers (c &d) is 1. Which means c+d = 2. This is only possible when c=d=1. And since they are in ascending order, this also means that a=b=1 too (no integer can be 0 since they are all positive integers). So a=b=c=d=1. This means e+f < 6 (total sum has to be less than 10). But we could have e=1, f=4 Or we could have e=2, f=3. Multiple possibilities for 'f'. Insufficient.

Combining the statements, we get that a=b=c=d=1. And now e+f=5. But again if e=1, f=4 and if e=2, f=3. WE cant uniquely determine value of 'f'. Insufficient.

Hi amanvermagmat, chetan2u

The question asks for the greatest value of the integer which can be only one value i.e. 4. Other possibilities are there but of all the possibilities greatest is 4. Can you please let me know if I am missing something.

Hi
you are mistaking it for 'greatest POSSIBLE'..
The Q is about greatest number and that will depend on the SUM, which can be anything<10
combined-
sum = $$\frac{3}{2}*6=9$$...
median is 1, so there are atleast 4 number of 1s..

numbers can be 1,1,1,1,1,4
or 1,1,1,1,2,3
so greatest could be 3 or 4
insuff
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If 6 positive integers have a sum less than 10, what is the value of   [#permalink] 11 Jan 2018, 07:24
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