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        riddles >> easy >> Murky Mergers
        
(Message started by: ThudanBlunder on Jan 25^th, 2009, 1:40am)

Title: Murky Mergers
Post by ThudanBlunder on Jan 25^th, 2009, 1:40am

A married couple open two separate market stalls. Jack sells only widgets at 2 for $1, having 90 in stock. Jill sells only gizmos at 3 for $2, also having 90 in stock. On the first day business is good and they sell all their stock for a combined turnover of $45 + $60 = $105. On the second day they decide to save money by renting only one stall and combining their respective stocks, but keeping the prices the same. As a result, instead of having to separately sell 2 widgets on Jack's stall and 3 gizmos on Jill's stall they can now wrap them up together and sell all 5 as one item for the same price of $3. Incredibly, at the end of the day they find that their turnover has increased to 3*(180/5) = $108!

Their two sons, with the same stock of widgets and gizmos, are selling them at 2 for $1 and 3 for $1 respectively, for a combined turnover of $45 + $30 = $75. Taking their parents' lead, they also decide to merge. However, they find that their combined stock sells for only 2*(180/5) = $72!

Their two daughters, who reckon it serves the brothers right for attempting to undercut their own parents, have the same stock and decide to sell it at the same price as Jack and Jill, except they try to shift more widgets by selling them at 6 for $3, rather than 2 for $1. Initially, their turnover was the same as their parents, $45 + $60 = $105. However, when they merged stock they found their combined turnover was just 5*(180/9) = $100! They couldn't understand how they could lose money when they had the same stock and sold it at the same price as their profitable parents. After all, the only thing they did differently was sell their 90 widgets in bigger boxes!

Go figure. ;D

Title: Re: Murky Mergers
Post by rmsgrey on Jan 25^th, 2009, 6:43am

So there's not a lot of difference between widgets and gizmos then?

Title: Re: Murky Mergers
Post by 0.999... on Jan 25^th, 2009, 7:27am

The two neighbors want to get in this business and set up separate stalls. The one sells 1 widget for $0.50 with a stock of 90, the other sells 3 gizmos for $2 with a stock of 90. This totals to $105 as before.
However, when the two combine, selling packages of 1 widget and 3 gizmos for $2.50, they make 2.5*(180/4) = $112.50!

Title: Re: Murky Mergers
Post by towr on Jan 25^th, 2009, 7:46am

The more widgets you sell as gizmos the better.
So bundle 0 widgets with 3 gizmos, and earn 2*(180/3)=$120.

Title: Re: Murky Mergers
Post by ThudanBlunder on Jan 25^th, 2009, 8:21am

on 01/25/09 at 07:27:50, 0.999... wrote:

However, when the two combine, selling packages of 1 widget and 3 gizmos for $2.50, they make 2.5*(180/4) = $112.50!

So what is the general rule for merging or not? :)

Title: Re: Murky Mergers
Post by ThudanBlunder on Jan 26^th, 2009, 11:45am

Let a/b and c/d be the original selling prices, where a/b > c/d > 0

Then the average price is
(ad + bc)/2bd before merging
and
(a + c)/(b + d) after merging.

So it will be better to merge when (a + c)/(b + d) > (ad + bc)/2bd
That is, when
(ad - bc)(b - d) > 0

The first term implies a/b > c/d, which WLOG has already been assumed.
So we can conclude that merging is advantageous only when
i) the prices are different
ii) the denominator of the higher price is greater than that of the lower price

The parents sold widgets for 1/2 (lower price) and gizmos for 2/3 (higher price) dollars each.
So merging was preferable as 3 > 2

The brothers sold widgets for 1/3 (lower price) and gizmos for 1/2 (higher price) dollars each.
So merging was not preferable as 2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ngeqslant.gif 3

The sisters sold widgets for 3/6 (lower price) and gizmos for 2/3 (higher price) dollars each.
So merging was not preferable as 3 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ngeqslant.gif 6

This means if you are selling stuff for a/b (higher price) and want to merge with a cousin who is selling stuff for c/d (lower price), it is a good idea to merge if b > d. However, if she decides to rewrite her price as nc/nd it is no longer a good idea if n > b/d. LOL

Title: Re: Murky Mergers
Post by Grimbal on Jan 26^th, 2009, 6:13pm

Simpler: If you don't mix up widgets and gizmos, but just group them, then the item with the larger denominator (or grouping size) will be left over. If the left-over items are of the lower price, merge to sell them at the average price. If the left-over items are of the higher price, don't merge so that they are sold at the highest price.

So the rule remains the same as ThudanBlunder's except in cases where the availability of each item differs. The comparison of b and d should be replaced by the comparison of b/m and d/n, where m and n are the initial stock or more expensive resp. less expensive items.