Last December, at Christmas Eve, Nagib called his three sons, Abdon, Fairuz and Samara, and gave them 90 turkeys to be sold at the fair.

- "Abdon, you will carry 10 turkeys. You, Fairuz, will carry 30 turkeys, and Samara the remaining 50 turkeys, and each one of you has to bring me $1,500 after selling all the turkeys.

Abdon is free to devise the strategy you all will use, that is, the price of one turkey established by Abdon for a group of turkeys must be exactly the same that you two have to follow.

Explaining better a two-step strategy: if Abdon decides to sell 4 of them by $1200 ($300 each), you two must sell any quantity you want, but by the same unit price ($300 each turkey). Then, if Abdon decides to sell the remaining 6 turkeys he has by $300 ($50 each), you two must sell your remaining turkeys by this same unit price ($50 each turkey).

But, I am not forcing that the strategy devised by Abdon must consist of only two steps. I gave you just an example which, by the way, doesn´t work for what I want. Moreover, in the strategy devised, in each step all three must sell at least one turkey, by an unitary price greater than zero."

What is the strategy devised by Abdon to insure that each one gathers exactly $1,500 for the turkeys they carried to the fair?

Initially, the most expensive bird will obviously be more than $150 (Adbon's average selling price), while the least expensive will be less than $30 (Samara's average price).  Adbon's sales strategy involves three rounds of selling.  He starts at $280 per turkey (YIKES!), then drops the price to $30 each, then again to $5 each.  The following are each brother's $1,500 in sales:

Adbon (10 birds):

5 @ $280 = $1,400

3 @ $30  = $90

2 @ $5 = $10 

Fairuz (30 birds):

3 @ $280 = $840

21 @ $30 = $630

6 @ $5 = $30

Samara (50 birds):

1 @ $280 = $280

39 @ $30 = $1,170

10 @ $5 = $50

Interestingly, the number of birds each sold in the first round (5/3/1) is the reverse of their original proportion of birds (1/3/5), while the numbers sold in round 3 follow the same proportions (2/6/10).  As well, the brother's total sales from rounds 1 and 3 each dropped consecutively by $540, while the round 2 sales in order obviously increased by the same amount.

Posted by rod hines on 2008-11-27 18:49:40