Mean/Average of Consecutive Numbers and Number Signs

[QuinganZ341]

If a DS questions tells me there are 5 consecutive even integers (no specific value provided) and ask me to provide the mean/average, I know I need to find the value of the middle number (3rd number). And if a statement provides me the sum of 2nd number plus 4th number, I know I can find the 3rd number by just dividing the sum by 2.

I don't understand, conceptually, why the aforementioned method works for both positive and negative numbers? I've tested cases with the 2nd number being a negative and the 4th number being a positive, and the mean always comes out to be same. So I know it's true - I just don't understand why. If you could point me to some gaps in my foundational knowledge, that would be great. Thanks much!

[tim]

Let's take your second example. If the integers are consecutive, the third number is exactly halfway between the second and fourth, so it is the average regardless of whether the numbers are negative, positive, zero, or some combination of those. Once you understand the definition of average, you can confidently apply it to every situation that requires it. That's one of the awesome things about math - once you have a rule, you KNOW it will apply to every situation that fits the rule. The average rule doesn't say "add the numbers and divide by how many IF they are all positive"; it just says "add the numbers and divide by how many". Once you have a rule and you understand it, you don't have to question why it works; it just works!

[QuinganZ341]

I see - thanks Tim!

[RonPurewal]

i'm lazier than tim when it comes to writing large numbers of words.

so:

* draw a few number lines.

* on those number lines, plot a few sets of 'consecutive even integers' (or whatever).

* look at them.

* note the spacing.

* it should become very clear, very quickly, why signs are irrelevant here.

[QuinganZ341]

i'm lazier than tim when it comes to writing large numbers of words.

so:

* draw a few number lines.

* on those number lines, plot a few sets of 'consecutive even integers' (or whatever).

* look at them.

* note the spacing.

* it should become very clear, very quickly, why signs are irrelevant here.

Did what you suggested and finally realized why I was confused:
- I never thought about averages in terms of negative numbers
- I only thought about negative numbers in terms of the space between them (absolute value)
- you find averages by adding two values
- you find the space by subtracting one value from another value
- thus, I didn't think about the fact that when I am adding two negative numbers/dividing by 2 - it's a value outcome, not a space outcome

[tim]

I don't think this is what happened to you, but I'd like to share a little observation:

I've had many students throughout the years who have gone to apply a theorem and gotten a result that didn't match what was supposed to happen, only to conclude that the theorem MUST be wrong because there's no WAY they could have been the one to make a minor calculation mistake. [It's kind of painful to watch - it's usually super gifted 9-year-olds with 200+ IQs for whom past experience has confirmed that when someone disagrees with them the other person is usually wrong. It's kind of a blow when they realize for the first time that they are fallible and that mathematical theorems are immune to second-guessing (UNlike science, I might note).] This is something Ron and I deal with quite a bit on the forums - smart GMAT students who get a little overconfident and want to start questioning official GMAT answers because they think they know better and don't take the time to figure out what sort of mistake THEY may have made.

Please note I'm not saying any of this applies to you at all; on the contrary, you came here telling us you KNOW the rule is correct and just want to understand WHY. This is commendable, and I wish more students would do this rather than just assuming they know more about the content of the GMAT than the creators of the GMAT. The only reason I write this rant here is to remind you to keep focusing on the right things, and to provide some guidance to other students. I could very easily see another student in a similar situation doing one of two things:

1) Convincing him/herself that negative numbers are "different" from positive numbers, so the rule "shouldn't work" for negatives. Students who fall into this trap are making a false distinction (without even testing its validity) that ends up causing them to reach a false conclusion.

2) Miscalculating the average (very easy to do especially when dealing with negatives and absolute values, and convincing him/herself that the rule therefore "must not work" for negatives. Students who fall into this trap should remember what I wrote above about the likelihood that the GMAT made a mistake versus the likelihood that they made a mistake (I'll give you a hint: it's 0%).

[RonPurewal]

well, it's not 0.0000000000 per cent. this is a GMAT PREP problem for which the given answer was, in fact, wrong:
https://www.manhattanprep.com/gmat/foru ... t6471.html

i don't know whether that problem has been fixed, or whether it's still in the software. but, as written, that problem should have answer E.

on the other hand, the reason i actually remember this problem is because it is THE ONLY TIME I HAVE EVER SEEN THIS HAPPEN.

[RonPurewal]

This is something Ron and I deal with quite a bit on the forums - smart GMAT students who get a little overconfident and want to start questioning official GMAT answers because they think they know better

yes, i'm always surprised by this ^^ ... but i'm especially surprised when it happens on a SC problem.

for instance, i'm a non-native speaker and writer of (among others) italian. if i were taking an italian-language SC exam—one that was written by, and meticulously edited by, professional italian writers and editors—i would never even THINK about trying to claim that the answers were 'wrong'.
(i would never even think about thinking about trying to claim that the answers were wrong.)
i mean... that's exactly the kind of thing for which the ancient greeks invented the word 'hubris'. hubris is a very very bad thing.

(also, amusingly enough, these 'criticisms' invariably come from posters whose own writing contains gross errors in almost every sentence. the internet has a name for this: "skitt's law".)

[RonPurewal]

anyway, we've wandered rather far from the original point of this thread, so... yeah. if you have further questions about the problem feel free to ask.

[tim]

well, it's not 0.0000000000 per cent. this is a GMAT PREP problem for which the given answer was, in fact, wrong:

Rounding error on my part.

You know, I just wish those same people who write grammatically atrocious posts about how much smarter they are than the GMAT would take your advice above to heart. But that would require them to overcome a particularly infectious type of hubris that is all too common on Internet forums. Trollers gonna troll I suppose.

NOW, to tie that error Ron caught back to the original topic of the thread, I'd like to encourage everyone to read that thread Ron linked to. If even someone working for the GMAT could make a mistake involving constraints on a problem, that should help all of you to see just how important it is to pay attention to whether numbers are positive or negative, integer or non-integer, and *more importantly* to pay attention when the GMAT *doesn't* say anything on the subject.