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M08-28

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M08-28 [#permalink]

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If the average of four distinct positive integers is 60, how many integers of these four are less than 50?


(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.

(2) The median of the four integers is 50.
[Reveal] Spoiler: OA

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Re M08-28 [#permalink]

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It's almost always better to express the average in terms of the sum: the average of four distinct positive integers is 60, means that the sum of four distinct positive integers is \(4*60=240\).

Say four integers are \(a\), \(b\), \(c\) and \(d\) so that \(0 \lt a \lt b \lt c \lt d\). So, we have that \(a+b+c+d=240\).

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190. The mdian of \(\{b,c,d\}\) is 51 means that \(c=51\). Now, if \(b=50\), then only \(a\), will be less than 50, but if \(b \lt 50\), then both \(a\) and \(b\), will be less than 50. But we are also given that \(c+d=190\). Substitute this value in the above equation: \(a+b+190=240\), which boils down to \(a+b=50\). Now, since given that all integers are positive then both \(a\) and \(b\) must be less than 50. Sufficient.

(2) The median of the four integers is 50. The median of a set with even number of terms is the average of two middle terms, so \(\text{median}=\frac{b+c}{2}=50\). Since given that \(b \lt c\) then \(b \lt 50 \lt c\), so both \(a\) and \(b\) are less than 50. Sufficient.


Answer: D
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Re: M08-28 [#permalink]

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In statement # 2:

What about the case when both 'b' and 'c' are 50?

The problem does not state b > c anywhere.

So we can have 2 valid set of elements:

Set 1: 1, 48, 52, 139 -> 2 elements are less than 50
Set 2: 1, 50, 50, 139 -> 1 element is less than 50.

Insufficient -> Answer must be A.
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Re: M08-28 [#permalink]

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Krabathor wrote:
In statement # 2:

What about the case when both 'b' and 'c' are 50?

The problem does not state b > c anywhere.

So we can have 2 valid set of elements:

Set 1: 1, 48, 52, 139 -> 2 elements are less than 50
Set 2: 1, 50, 50, 139 -> 1 element is less than 50.

Insufficient -> Answer must be A.


Please read the question carefully: If the average of four distinct positive integers is 60...
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Re: M08-28 [#permalink]

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New post 11 Feb 2016, 08:02
a1+a2+a3+a4=240
Statement 1:
a3=51
a3+a4=194
a4=143
so a1+a2=46 so both ae less than 50

Statement 2
a2+a3/2=50
a2+a3=100
a1+a4=140
since a2 and a3 are taken for median so a1 is less than a2 and a3.Hence less than 50
Sufficient

So D
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Re: M08-28 [#permalink]

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New post 04 Jun 2016, 23:01
Bunuel wrote:
If the average of four distinct positive integers is 60, how many integers of these four are less than 50?


(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.

(2) The median of the four integers is 50.


Dear bunuel,

Can you post more of such questions??
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Re: M08-28 [#permalink]

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New post 05 Jun 2016, 03:06
Ashishsteag wrote:
Bunuel wrote:
If the average of four distinct positive integers is 60, how many integers of these four are less than 50?


(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.

(2) The median of the four integers is 50.


Dear bunuel,

Can you post more of such questions??


Search in Statistics and Sets problems: search.php?search_id=tag&tag_id=34
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Re M08-28 [#permalink]

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New post 02 Jul 2016, 22:44
I think this is a high-quality question and I agree with explanation.
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Re M08-28 [#permalink]

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New post 06 Jul 2016, 04:06
I think this is a high-quality question and I agree with explanation.
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Re: M08-28 [#permalink]

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New post 02 Oct 2016, 19:46
I don't know if I'm reading the same question that the explanation...

If the average of four distinct positive integers is 60, how many integers of these four are less than 50?
(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.
(2) The median of the four integers is 50.

(1) Sufficient --> 'a','b' and 'c'> 50 so 3 integers are less than 50.
(2) Insufficient-> 'a','b' are less than 50, 'c' could be more o less than 50. So answer could be 2 or 3

Can anyone explain me what I'm missing?
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Re: M08-28 [#permalink]

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New post 03 Oct 2016, 06:42
GiancarloGV wrote:
I don't know if I'm reading the same question that the explanation...

If the average of four distinct positive integers is 60, how many integers of these four are less than 50?
(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.
(2) The median of the four integers is 50.

(1) Sufficient --> 'a','b' and 'c'> 50 so 3 integers are less than 50.
(2) Insufficient-> 'a','b' are less than 50, 'c' could be more o less than 50. So answer could be 2 or 3

Can anyone explain me what I'm missing?


(2) The median of the four integers is 50. The median of a set with even number of terms is the average of two middle terms, so \(\text{median}=\frac{b+c}{2}=50\). Since given that \(b \lt c\) then \(b \lt 50 \lt c\), so both \(a\) and \(b\) are less than 50. Sufficient.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: M08-28 [#permalink]

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New post 11 Feb 2018, 19:42
Bunuel

What if the numbers are

x1,x2,x3,x4 where x2=50 and x3=51?

Then clearly (2) would be insufficient.

Could you please clarify.


Thanks,

ucb2k7
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Re: M08-28 [#permalink]

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Re: M08-28   [#permalink] 11 Feb 2018, 20:58
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