Grouping Qs: Word Translations, 5th Edition (page 81 online)

[BernardK777]

With regards to Grouping Problems in the Extra Problem Types Chapter, the book says:

"For some problems, may want to think about the most or least evenly distributed arrangement of the items. That is, assign to groups as evenly (or unevenly) as possible to create extreme cases."

Can you provide a problem and explanation of that problem in which this strategy is used because I am having trouble understanding it.

Thanks.

[RonPurewal]

Let's say your store sold 50 items over four days, selling a different number of items each day.
Most even distribution: 11, 12, 13, 14.
Least even distribution: 0, 1, 2, 47.

That kind of thing.

[RonPurewal]

Can you provide a problem and explanation of that problem in which this strategy is used because I am having trouble understanding it.

Thanks.

Does the book just throw this at you, without a problem to serve as an illustration?
If it does... well, that's really weird.

Ok, I'll make something up. Try to solve the problem before you scroll down to the next post.

A store sold a total of 50 items over a certain four-day period. Was there at least one day on which the store sold 14 or more items?

(1) The greatest number of items sold by the store on any one of the four days was 3 more than the least number of items sold by the store on any one of the four days.

(2) The store did not sell the same number of items on any two of the four days.

[RonPurewal]

That problem is definitely not happening with algebra, so we'll have to test cases.
As always when testing cases, try for "not sufficient". In other words, try to find...
... a case in which ≥14 items were sold on one day,
... a case in which, every day, ≤13 items were sold.

The "border" cases (for the greatest # of items sold in a day) are 13 and 14, so those are the cases of greatest interest to us here. (No sense trying to get a number bigger than 14 or less than 13 if we can get those.)

Statement 1: The greatest and least numbers of items are 3 apart.

Ok. If the greatest number is 14, the least number must be 11. Can we do that?
Well, there are 25 items left. So... sure. 14, 14, 11, 11 works. So does 14, 13, 12, 11.

If the greatest number is only 13, then the smallest number must be 10.
Can we do that?
50 - 13 - 10 = 27 items left.
Won't work"”"”you can't split 27 into two numbers that are both 13 or less.

Clearly won't work with any number smaller than 13, either"”"”you'll just have the same problem, only worse. So, sufficient.

--

[RonPurewal]

Statement 2: You need 4 different numbers.
This is where the "most spread out" / "least spread out" thing comes into play.

The MOST spread-out distribution is 0, 1, 2, 47. So it's clearly possible to have ≥14 items in a day.

To see whether we can get ≤13 items in a day, "crunch" the numbers together as much as possible.
"Closest together" is the way to make the SMALLEST number AS BIG as possible ... or to make the BIGGEST number AS SMALL as possible. That's why we're doing it here"”"”and, I imagine, why you'd do it anywhere else.
The average number of items is 50/4 = 12.5 per day, so the "closest scrunched together" solution will be wrapped around that average: 11, 12, 13, 14.
That's the lowest you can go with the maximum. If you make any number in this list smaller, you'll have to make the max bigger.

So, you're stuck with ≥14, so, sufficient again. It's D.

[RonPurewal]

By the way"”"”If you have a total of 50 books WITHOUT either of these restrictions, then it's possible to get both cases.

E.g., with 11, 12, 13, 14, you have a day with ≥14 items sold.
With 12, 12, 13, 13, or with 11, 13, 13, 13, you don't. (These are the only two groups of numbers in which you don't.)

As far as your original question, that is probably addressed most directly by the boldface thingy in the last post.

[BernardK777]

Ron,

Thank you for your responses. They were very helpful. And to answer your question regarding the book, no problem / example was offered as an illustration.

[tim]

Sorry to hear the book didn't provide examples. I'm glad you found Ron's posts helpful.

[RonPurewal]

Ron,

Thank you for your responses. They were very helpful. And to answer your question regarding the book, no problem / example was offered as an illustration.

Well, that's weird.

Now you have an example. Problem fixed (for you, at least).