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02 Oct 2018, 04:52
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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.
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
!
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
!
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Thank you for understanding, and happy exploring!
02 Oct 2018, 05:20
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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!
07 Oct 2018, 12:52
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gmat800live
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.
