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If the average (arithmetic mean) of 5 positive temperatures

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If the average (arithmetic mean) of 5 positive temperatures  [#permalink]

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New post 06 Mar 2011, 09:31
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If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be:

A. 6x
B. 4x
C. 5x/3
D. 3x/2
E. 3x/5
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Re: temperatures  [#permalink]

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New post 07 Mar 2011, 03:09
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5
Madelaine88 wrote:
If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be:
A/ 6x
B/ 4x
C/ 5x/3
D/ 3x/2
E/ 3x/5


Note that we have 5 positive temperatures.

Next, as the average is x then the sum of the temperatures is 5x and as all the temperatures are positive then the sum of the 3 greatest must be more than (or equal to) 3x (as the average of the 3 greatest must be at least x) and less than 5x: 3x<SUM<5x --> only option B fits.

Answer: B.
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Re: temperatures  [#permalink]

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New post 06 Mar 2011, 09:48
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The sum of three greatest should be more than sum of two lowest.

The total sum is; 5x

A. 6x; 6x is more than 5x. Not possible.
B. 4x; 5x-4x=x(Possible)
C. 5x/3; 10x/3; 10x/3 > 5x/3. Not possible
D. 3x/2; 7x/2; 7x/2 > 3x/2. Not possible
E. 3x/5; 22x/5; 22x/5 > 3x/5. Not possible.

Ans: "B"
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Re: temperatures  [#permalink]

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New post 06 Mar 2011, 23:55
I took 5 numbers and tried it out, because it's a "could be", so a possibility should be enough.

1, 2, 3, 4, 5 and mean = 15/5 = 3

so highest 3 = 3 + 4 + 5 = 12 = 4*3

So answer is B.

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Re: temperatures  [#permalink]

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New post 07 Mar 2011, 21:25
Temperatures should be positive.
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Re: If the average (arithmetic mean) of 5 positive temperatures  [#permalink]

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New post 06 Oct 2013, 21:50
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Madelaine88 wrote:
If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degrees Fahrenheit, could be:

A. 6x
B. 4x
C. 5x/3
D. 3x/2
E. 3x/5


Responding to a pm:

Avg of 5 temperatures is x. So the sum of all 5 temperatures is 5x.
Now what CAN be the sum of the 3 greatest temperatures?

Let's try to find the maximum value that the 'sum of the 3 greatest temperatures' can take and the minimum value that it can take.

Maximum: To make the sum of 3 greatest temperatures as large as possible, we make the 2 lowest temperatures as small as possible. The two lowest temperatures can be as small as 0.0000000000000001 i.e. anything slightly more than 0. So the sum of the 3 greatest temperatures will be slightly less than 5x.

Minimize: To minimize the sum of 3 greatest temperatures, we make the 2 lowest temperatures as high as possible. For the average to be x, either some values should be less than x and some more OR all values could be equal to x. That is, the temperatures could be x, x, x, x, x - in this case the two lowest temperatures are maximum (all temperatures are the same actually). So sum of 3 greatest temperatures will be at least 3x.

Note that only one value lies between 5x and 3x and that is 4x.

We don't really need to figure out the minimum sum. Once we know that maximum sum can be a little less than 5x, we see that the sum of 3 greatest temperatures can easily be 4x. We will be left with a sum of x for the two lowest temperatures. They can be x/2 each. The 3 greatest temperatures can be x, x and 2x or many other variations.
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Re: If the average (arithmetic mean) of 5 positive temperatures  [#permalink]

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New post 06 Oct 2013, 21:52
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Re: If the average (arithmetic mean) of 5 positive temperatures  [#permalink]

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New post 05 Dec 2015, 11:26
Hi All,

One of the great 'shortcuts' to this question is that it asks for what COULD be the sum of the three highest temperatures. As such, once you find an answer that COULD be the sum, you can STOP working. Once you prove that Answer is a possibility, then you've answered the question (and you can stop).

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Re: If the average (arithmetic mean) of 5 positive temperatures  [#permalink]

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New post 26 Sep 2016, 22:55
let x be 2 and total sum becomes 10
now using maximization and minimization the least values can be 1 and 1 equaling to 2 =>10-2 = 8 (rest of the temp)

substituting x in the options gives the answer B
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Re: If the average (arithmetic mean) of 5 positive temperatures  [#permalink]

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New post 17 Dec 2016, 21:46
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Re: If the average (arithmetic mean) of 5 positive temperatures  [#permalink]

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New post 02 Mar 2018, 11:19
Hi All,

This question has a couple of great "concept" shortcuts built into it.

We're told that the average of 5 positive numbers is X, which means that the SUM of the 5 numbers is 5X.

The question asks us for a POSSIBLE sum of the greatest 3 values….

Since the values are all POSITIVE and they sum to 5X, there's no way for the sum of the greatest 3 to be bigger than that. 6X is NOT possible. Eliminate A.

IF all the values were the same (just call them all X), then the biggest 3 would sum to 3X. Changing the values would only INCREASE the sum of the biggest 3. This helps to eliminate C, D and E - they're all too small.

There's only one answer left…

Final Answer:

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Re: If the average (arithmetic mean) of 5 positive temperatures &nbs [#permalink] 02 Mar 2018, 11:19
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