If the average (arithmetic mean) of 5 positive temperatures

[guest2]

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) 35x
(D) 23x
(E) 53x

The answer is B. But I don't quite understand how it could be 4x.
I started out by writing average = sum/n
which in this case leads to x = sum/5 so the sum = 5x

So I ruled out A since that would be larger than the sum of all 5 numbers. But what do I do next?

[divya]

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) 35x
(D) 23x
(E) 53x

so we know the average is x.
number of terms = 5
sum = 5x

since these are positive temps, no negative and zeros, the sum of the greatest 3 has to be less than 5x , also x has to be positive.

If you add the last 3 positive ##s the sum should be < sum of all 5
the only answer choice that fits is 4x, i.e B
all others choices are greater than 5x

[RonPurewal]

So I ruled out A since that would be larger than the sum of all 5 numbers. But what do I do next?

next, you rule out (c), (d), and (e), for exactly the same reason you ruled out (a).
that leaves (b).
done.

[sudaif]

The questions answer choices here are pasted incorrectly
A) 6x
B) 4x
c) 5x/3
d) 3x/2
e) 3x/5

Answer is B. But not sure how they arrived at it.

Instructors, pls advise.

[rohit801]

The questions answer choices here are pasted incorrectly
A) 6x
B) 4x
c) 5x/3
d) 3x/2
e) 3x/5

The Sum of the 3 GREATEST numbers CAN'T be LESS than the sum of 2 remaining numbers. So, if the total is 5x, and the sum of 3 greatest numbers is choice C,D, or E, then Sum of the remaining 2 numbers, i.e., 5x-(choice C,D, or E] is Greater than
the sum of 3 greatest numbers - not possible.

[vinayak.stalwart]

Why are we rejecting the option A outrightly ? It could be that the the two lowest temperatures are in negative ( since Fahrenheit can be -ve as well ) and in that case the sum of three greatest temperatures could exceed 5x.

Thanks,
VK

[sudaif]

the question says "positive temperatues"
can the instructors please answer the question with the correctly pasted answers? thank you.

[RonPurewal]

the question says "positive temperatues"
can the instructors please answer the question with the correctly pasted answers? thank you.

sure.

first, let's recount the facts that we know:
* the SUM of all five temperatures is 5x.
* all of the temperatures are positive.

6x is wrong because it's greater than the sum of all of the temps. that would be possible if some of the temperatures were allowed to be negative (so that you can add up some of them, and get a greater sum than you would by adding all of them), but it's not possible if the temperatures must be positive.

now:

we're looking at the three GREATEST temps. the three greatest temperatures must have an average of at least x (it's impossible for the three greatest values to have an average that is lower than the average for all the values).
therefore, the three desired values must add up to at least 3x.

this observation rules out the last three choices.

--

you can also solve this problem by plugging in a specific number, such as 10, for x.
then you have 5 positive temps that add to 50.

choice (b) is 40, which can be done if your temps are, say, 5, 5, 10, 15, and 15.
since this is a "could" question, you are done as soon as you find any such possibility (there is nothing special about this particular one).

[sudaif]

the question says "positive temperatues"
can the instructors please answer the question with the correctly pasted answers? thank you.

), but it's not possible if the temperatures must be positive.

sure.

first, let's recount the facts that we know:
* the SUM of all five temperatures is 5x.
* all of the temperatures are positive.

6x is wrong because it's greater than the sum of all of the temps. that would be possible if some of the temperatures were allowed to be negative (so that you can add up some of them, and get a greater sum than you would by adding all of them

now:

we're looking at the three GREATEST temps. the three greatest temperatures must have an average of at least x (it's impossible for the three greatest values to have an average that is lower than the average for all the values).
therefore, the three desired values must add up to at least 3x.

this observation rules out the last three choices.
.

Basically, the remainder when subtracted by 5x MUST be less than the sum of the three greatest. You said it more elegantly, i.e. the three greatest, each cannot be less than the overall average. they must at least be equal to the overall average -- otherwise, the average could never be x.

thanks Ron!!

[tim]

:)

[ers92]

the sum of the three greatest integers must be at least 3x, ok. But answer C (5x/3) is > 3x. How can you rule this one out?

Thanks

Er

[tim]

You lost me at the part where you claimed that 5x/3 > 3x. x has to be positive based on the constraints of the problem, so 5x/3 is definitely less than 3x.

[ers92]

Sorry, 5x/3 is less than 3x. You're right :D

[RonPurewal]

Sorry, 5x/3 is less than 3x. You're right :D

note that, if non-positive temperature values (zero or negatives) were allowed, this would actually be a non-trivial issue.

those values aren't allowed in this problem, but one should always double-check.