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Manager  Joined: 23 Jan 2006
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If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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If the average (arithmetic mean) of four different numbers is 30, how many of the numbers are greater than 30?

(1) None of the four numbers is greater than 60.
(2) Two of the four numbers are 9 and 10, respectively.

Originally posted by kook44 on 22 Jun 2006, 18:13.
Last edited by Bunuel on 06 Jun 2019, 02:45, edited 3 times in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58435
Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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artugoca wrote:
I have seen a lot of post in this question that are considering same numbers to proof the statement wrong. May be I am wrong but aren't we supposed to consider different numbers?

Thanks

Yes, the numbers should be distinct though the answer still remains C.

If the average (arithmetic mean) of four different numbers is 30, how many of the numbers are greater than 30?

$$a+b+c+d=4*30=120$$

(1) None of the four numbers is greater than 60. Many combinations are possible. For example, numbers can be: 20-25-30-45 (only one number is greater than 30) OR 15-20-40-45 (two number are greater than 30). Not sufficient.

(2) Two of the four numbers are 9 and 10 respectively. Also not sufficient, consider: 0-9-10-101 OR 9-10-35-66. Not sufficient.

(1)+(2) As two of the four numbers are 9 and 10 then the sum of other two must be 120-(9+10)=101. Now, as the greatest number can be at most 60, then the least value of the other one is 41, so in any case two numbers will be more than 30. Sufficient.

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VP  Joined: 02 Jun 2006
Posts: 1016
Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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1
Let x1, x2, x3, x4 be the #s

Given average = 30 i.e. sum = 120

S1: All #s < 60
Can come up with two sets {10,10, 50, 50} or {35, 35, 35, 15} which has 2 different answers (2 & 3).
Not sufficient.

S2: x1 = 9, x2 = 10
x3+x4 = 120-19 = 101
Possible Values :
x3 = 100, x4 = 1 OR
x3 = 50, x4 = 51
Two different answers. Not sufficient.

S1 & S2:
x3+x4 = 101
Max. possible value for {x3, x4} = 59
As sum has to be 101, other value has to be 41
Therefore enough to answer question: 2

Sufficient.

Intern  Joined: 17 Oct 2013
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Concentration: Entrepreneurship, General Management
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Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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I have seen a lot of post in this question that are considering same numbers to proof the statement wrong. May be I am wrong but aren't we supposed to consider different numbers?

Thanks
Intern  Joined: 04 Feb 2015
Posts: 1
Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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Hi Guys,

could one of you possibly explain to me why the from the two statements we are certain of the number of values greater than 30. From what I understand we have 101 left to a and d so couldnt a=1 and d=101?

Really appreciate the help - thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 58435
Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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Freddy123 wrote:
Hi Guys,

could one of you possibly explain to me why the from the two statements we are certain of the number of values greater than 30. From what I understand we have 101 left to a and d so couldnt a=1 and d=101?

Really appreciate the help - thanks

Have you read this: if-the-average-arithmetic-mean-of-four-different-numbers-is-100604.html#p1322958 ?

As two of the four numbers are 9 and 10 then the sum of other two must be 120-(9+10)=101. Now, as the greatest number can be at most 60 (from 1), then the least value of the other one is 41, so in any case two numbers will be more than 30.
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Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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Freddy123 wrote:
Hi Guys,

could one of you possibly explain to me why the from the two statements we are certain of the number of values greater than 30. From what I understand we have 101 left to a and d so couldnt a=1 and d=101?

Really appreciate the help - thanks

Another solution is to use the data points of the average and their particular differentials:

Test (1)
• Not sufficient because it is obvious: When you choose 30, 30, 30, 30 four numbers equal 30 and when you choose 10,10,50,50 two numbers are over 30

Test (2)
• Data Point 9 leads to differential "-21"
• Data Point 10 leads to differential "-20"
• To balance these underweight you can choose
• 51 (30+21) and 50 (30+20) as additional data points or
• 71 (30+41) and 30 (30+0; no difference to the average) as additional data points
• that is, not sufficient

Test (1) & (2)
• Tells you that you cannot choose 71 and 0 because (1) says every number is less than 60. Hence you can choose only the following data points
• 51 and 50 (as described above)
• 60 (30 + 30) and 41 (30 + 11). This solution shows the maximum of one data point (60)
Manager  B
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Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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alexanderbell wrote:
Freddy123 wrote:
Hi Guys,

could one of you possibly explain to me why the from the two statements we are certain of the number of values greater than 30. From what I understand we have 101 left to a and d so couldnt a=1 and d=101?

Really appreciate the help - thanks

Another solution is to use the data points of the average and their particular differentials:

Test (1)
• Not sufficient because it is obvious: When you choose 30, 30, 30, 30 four numbers equal 30 and when you choose 10,10,50,50 two numbers are over 30

Test (2)
• Data Point 9 leads to differential "-21"
• Data Point 10 leads to differential "-20"
• To balance these underweight you can choose
• 51 (30+21) and 50 (30+20) as additional data points or
• 71 (30+41) and 30 (30+0; no difference to the average) as additional data points
• that is, not sufficient

Test (1) & (2)
• Tells you that you cannot choose 71 and 0 because (1) says every number is less than 60. Hence you can choose only the following data points
• 51 and 50 (as described above)
• 60 (30 + 30) and 41 (30 + 11). This solution shows the maximum of one data point (60)

You can't pick the same numbers because one of the conditions in this question is that the each number is different. The answer remains C in this question but keep an eye out for this detail for future questions.
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Re: If the average (arithmetic mean) of four different numbers is 30, how  [#permalink]

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_________________ Re: If the average (arithmetic mean) of four different numbers is 30, how   [#permalink] 06 Jun 2019, 02:44
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