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Intern  B
Joined: 06 Sep 2018
Posts: 36
GMAT 1: 760 Q50 V44 GMAT 2: 740 Q48 V44 Interesting set/average conundrum - Need expert insight  [#permalink]

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Hi all! Imagine I give you a set with N number of terms, and I tell you that the Avg of set is = A, and I tell you that the range of the set is R. Now I ask you, what is the highest value (B) that one of the terms of the set can have?

Is it correct to say that A <= B < A + R? When R is 0 though, the biggest number is of course A. Now, I think the relationship mentioned works for positive numbers, but not 100% sure.

For instance, take a set with A = 20, 4 terms, and Range 2. According to formula above, the biggest number has to be less than 22. Let's check -> {20,20,20,20} which becomes {19,20,20,21} when we set the range. Now you can basically add to lowest and add to the highest or subtract to lowest and subtract from highest to maintain the A. In this case we can see that there's no way we can go higher than 22. We can go to 21.5 but not 22.

We can put another example. Set with A = 5, 10 terms, Range 4. Biggest number has to be less than 9.
{5,5,5,5,5,5,5,5,5,5} which becomes {3,5,5,5,5,5,5,5,5,7} when we set the range. And now proceed to add to lowest and highest. {4,5,5......,5,8}. And you can see that biggest number will be 8.999999 or 8 if we are talking about integers.

So, my question is. Is there any rule, or any process or anything that dictates the behavior above? I am really interested in understanding in depth how constraints to sets matter especially on maximizing and minimizing problems. Thanks so much everyone and ghnlrug for exploring these concepts together :D!
Intern  B
Joined: 06 Sep 2018
Posts: 36
GMAT 1: 760 Q50 V44 GMAT 2: 740 Q48 V44 Re: Interesting set/average conundrum - Need expert insight  [#permalink]

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Found one case that breaks the lower boundary though. If you don't have enough terms to re-deliver value, then B > A is not enough. Set of 3 numbers, average 15, range 4 for instance. {15,15,15} -> {13,15,17} Now... let's try biggest number 16... then {12,16,16} and the sum is now 44. We are missing 1 that we can't allocate anywhere... so yeah... I'd love if anyone can help understand this better. I have not yet found cases where the top boundary breaks though!
Manhattan Prep Instructor G
Joined: 04 Dec 2015
Posts: 832
GMAT 1: 790 Q51 V49 GRE 1: Q170 V170 Re: Interesting set/average conundrum - Need expert insight  [#permalink]

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gmat800live wrote:
Hi all! Imagine I give you a set with N number of terms, and I tell you that the Avg of set is = A, and I tell you that the range of the set is R. Now I ask you, what is the highest value (B) that one of the terms of the set can have?

Is it correct to say that A <= B < A + R? When R is 0 though, the biggest number is of course A. Now, I think the relationship mentioned works for positive numbers, but not 100% sure.

For instance, take a set with A = 20, 4 terms, and Range 2. According to formula above, the biggest number has to be less than 22. Let's check -> {20,20,20,20} which becomes {19,20,20,21} when we set the range. Now you can basically add to lowest and add to the highest or subtract to lowest and subtract from highest to maintain the A. In this case we can see that there's no way we can go higher than 22. We can go to 21.5 but not 22.

We can put another example. Set with A = 5, 10 terms, Range 4. Biggest number has to be less than 9.
{5,5,5,5,5,5,5,5,5,5} which becomes {3,5,5,5,5,5,5,5,5,7} when we set the range. And now proceed to add to lowest and highest. {4,5,5......,5,8}. And you can see that biggest number will be 8.999999 or 8 if we are talking about integers.

So, my question is. Is there any rule, or any process or anything that dictates the behavior above? I am really interested in understanding in depth how constraints to sets matter especially on maximizing and minimizing problems. Thanks so much everyone and ghnlrug for exploring these concepts together :D!

Just to make sure I understand, let me rephrase what you're saying:

"Suppose that you have a set of numbers, and you know their average and their range. The largest possible number that can be in that set has to be at least as big as the average, and no bigger than the average + the range."

If that's what you're saying, then it's correct. Here's the reasoning behind it.

- If the largest number in a set was smaller than the average of that set, then every number in the set would have to be smaller than the average of the set. That's impossible - for instance, you'll never average 1, 2, 3, 4, and 5 and end up with 10 (unless you made a math mistake). The average is always somewhere in the 'middle' of the set - it doesn't have to be in the exact middle, but it does have to be somewhere within the range of the set.

- If the largest number in a set was larger than average+range, here's what that would imply. The range is the largest number minus the smallest number.

Range = largest - smallest
largest = smallest + range
largest > average+range
smallest+range > average+range
smallest>average

So, the smallest number in the set would have to be higher than the average. So, every number in the set would have to be higher than the average. That's also impossible, for the reason described above. You can't average (for example) 100,200,300, and 400 and expect to get an answer like 20 or 50!

- Since both of those scenarios are impossible, we know that the largest number has to be somewhere between those two bounds. It has to be at least as big as the average, but no bigger than average+range.
_________________ Chelsey Cooley | Manhattan Prep | Seattle and Online

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Intern  B
Joined: 06 Sep 2018
Posts: 36
GMAT 1: 760 Q50 V44 GMAT 2: 740 Q48 V44 Re: Interesting set/average conundrum - Need expert insight  [#permalink]

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ccooley I'm so grateful for your help here. Ok, so now that I understand that for sure if largest number is L, average is A and range is R, then A<= L < A+R now my question is... why isn't there a combination of 3 terms that give me a set of range 4 and average 15, where the highest number is a 16?

According to formula above, for such set the largest number should be equal or more than 15 an less than 19. The number 16 meets that inequality, but I can't find a way to make it the largest term while keeping range at 4 and avg 15.

Any insight will be appreciated!
Math Expert V
Joined: 02 Aug 2009
Posts: 7978
Re: Interesting set/average conundrum - Need expert insight  [#permalink]

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gmat800live wrote:
ccooley I'm so grateful for your help here. Ok, so now that I understand that for sure if largest number is L, average is A and range is R, then A<= L < A+R now my question is... why isn't there a combination of 3 terms that give me a set of range 4 and average 15, where the highest number is a 16?

According to formula above, for such set the largest number should be equal or more than 15 an less than 19. The number 16 meets that inequality, but I can't find a way to make it the largest term while keeping range at 4 and avg 15.

Any insight will be appreciated!

Hi..
You will not get this because moment you make 16 as largest and take range 4 with average 15, the largest 16 will no longer remain the largest..
Say 16 and range 4 is possible so two numbers are 12 and 16.
Third number = 15*3-12-16=45-28=17

Why? Because 16 gives you ONE more than the average, so other two COMBINED can be ONE less than average..
_________________ Re: Interesting set/average conundrum - Need expert insight   [#permalink] 08 Oct 2018, 05:25
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