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Positive integers a, b, c, d and e are such that a<b<c<d<e [#permalink]
18 Oct 2010, 09:57

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Positive integers a, b, c, d and e are such that a < b < c < d < e. If the average (arithmetic mean) of the five numbers is 6 and d - b = 3, then what is the greatest possible range of the five numbers?

Positive integers a, b, c, d and e are such that a < b < c < d < e. If the average (arithmetic mean) of the five numbers is 6 and d - b = 3, then what is the greatest possible range of the five numbers?

12 17 18 19 20

Range = e-a. d-b=3 or d=b+3

Average = 6 thus a+b+c+d+e=30

To maximize range, we need to minimize a and maximize e. Minimum a is 1, so we choose that. Since sum is fixed, we should choose minimum possible values for b,c,d to maximize e. b>a, minimum choice is 2 c>b, minimum choice is 3 d=b+3 so d=5 So e(max) = 30 - a(min) -b(min) -c(min) - d(min) = 30-1-2-3-5 = 19 Range(max) = e(max)-a(min) = 19-1 =18

Well it's basic logic. If you want maximize one of the numbers you should minimize all other numbers if you're given the average or sum of the series. An example would be there are five numbers the sum of the five numbers add to 20. All the numbers are positive what is the largest possible number in the set. The smallest positive number is 1. 20-1-1-1-1 =16. If other numbers were not minimize then the answer would be less than 16. 20-1-2-1-2 = 14.

For the above question, the smallest positive integer is 1. Therefore A=1. Since they can't be the same number the next smallest integer is 2. B = 2. We know D-B = 3, so 2+3 = 5. D=5. C has to be between 2 and 5. 3 is the smallest number. Finally the last number should be 30-5-3-2-1=19.

Let's say you didn't choose the smallest number a = 3, then b = 4, c = 5, d = 7. E would be only be 11. The range would only be 8.

Re: Positive integers a, b, c, d and e are such that a<b<c<d<e [#permalink]
18 Sep 2013, 06:31

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Re: Positive integers a, b, c, d and e are such that a<b<c<d<e [#permalink]
30 Sep 2013, 10:24

a<b<c<d<e

As all these integers are +ve then minimum value of a=1.

Furthermore, to maximize the range we will have to minimize the value of "a" and maximize the value of "e".

a=1 b=2 c=3 d= 5 (Since d=b+3) e=19

Range = e - a = 19 -1 Range = 18 ! _________________

Rgds, TGC! _____________________________________________________________________ I Assisted You => KUDOS Please _____________________________________________________________________________

Re: Positive integers a, b, c, d and e are such that a<b<c<d<e [#permalink]
17 Feb 2014, 06:51

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My way:: let's say that 'a' the smallest integer is 'x' and then we have that b=x+1, c=x+2, d=x+4 and e=x+k. We then have that the sum 5x+7+k = 30, and that k = 23-5x. Since the smallest value of x can be 1. Therefore k = 18 which is also the range is the maximum value. C is the correct answer

Positive integers a, b, c, d and e are such that a<b<c<d<e [#permalink]
02 Jul 2014, 03:42

shrouded1 wrote:

shrive555 wrote:

Positive integers a, b, c, d and e are such that a < b < c < d < e. If the average (arithmetic mean) of the five numbers is 6 and d - b = 3, then what is the greatest possible range of the five numbers?

12 17 18 19 20

Range = e-a. d-b=3 or d=b+3

Average = 6 thus a+b+c+d+e=30

To maximize range, we need to minimize a and maximize e. Minimum a is 1, so we choose that. Since sum is fixed, we should choose minimum possible values for b,c,d to maximize e. b>a, minimum choice is 2 c>b, minimum choice is 3 d=b+3 so d=5 So e(max) = 30 - a(min) -b(min) -c(min) - d(min) = 30-1-2-3-5 = 19 Range(max) = e(max)-a(min) = 19-1 =18

Answer is (c)

I have some inference like you, but

a + b + c + d + e = 30 d-b =3 => a + 2b + c + e = 27 => e = 27 - a - 2b - c range = e - a = 27 - 2a - 2b - c

range is max when a, b, e are min => a=1, b=2, c=3 (as a,b,c are positive integers and a<b<c)

Re: Positive integers a, b, c, d and e are such that a<b<c<d<e [#permalink]
07 Jul 2015, 23:01

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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