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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Karishma
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