Math 115                                             Name:                          Key                                                     

Quiz 3                                                  February 20, 2013                                          

 

1.  Andy and Terry split a cake by the divider-chooser method.  Andy, the divider, likes chocolate icing twice as much as vanilla.  Show 2 different ways that Andy might split the cake to ensure he gets a “fair” piece. 

 

One way to divide it fairly is to slice it into 2 identical parts: horizontally, through the middle, so each person will get 2 chocolate and 2 vanilla pieces.

The other fair way to divide it is to make one part contain 3 chocolate pieces, and one part contain all 4 vanilla and 1 chocolate.  As explained in class, this can be seen to be fair in a couple of ways, including: if vanilla pieces are worth 1 (quarter, dollar, or whatever) then chocolate pieces are worth 2 (twice as much), and each part is now worth 6.  Alternatively, if you start with 2 equal parts, as in the first solution, and then swap 1 chocolate piece for 2 vanillas, which is a fair swap to Andy, you wind up with this 2nd solution.


2.  Three players, Ali, Bo, and Caz, divide the 13 items shown by the method of markers. 

1       2      3         4        5       6      7        8        9       10     11    12   13

 
 


Which items: go to Ali?   9-13

                      go to Bo?   1-3

          go to Caz?  6-7

                      are (initially) left over?  4,5,8

 

2b.  According to the rules of this method, how should the leftovers be allocated?  EXPLAIN your answer in a complete English sentence.

It doesn’t matter, as each players has already received a share that they consider fair.  At this point each could receive anything additional (or nothing additional) and it would still be fair.

 

 

 

 

Adam

Bob

Chris

Games

30

36

29

Clothes

18

15

16

3.  3 brothers, Adam, Bob, and Chris, divide some games and some clothes by the method of sealed bids. 

The bids (in $) are as follows: 

a.        Find the value of each brother’s fair share

Adam:  1/3 of $48 = $16

Bob:  1/3 of $51 = $17

Chris: 1/3 of $45 = $15

 

 

b.      Describe the final settlement (what each brother gets/pays after the first settlement and any surplus is distributed)

 

First, Adam gets the clothes (worth $18), and Bob gets the games (worth $36).  Since Adam received $2 more in value than his fair share, he pays $2 in cash.  Bob received $19 in value more than his fair share, so he pays $19 in cash.  Chris, who received nothing, takes his fair share of $15 in cash from the money that Adam and Bob paid.  There is now $6 leftover ($21 paid, $15 claimed), so this can be split evenly, giving each brother $2.  So in the end Adam gets the clothes and neither pays nor receives any money (he paid $2 but then got $2 back); Bob got the games and paid $17 (paid $19 before getting $2 back), and Chris received $17 cash ($15+2).  Notice that the cash payments all exactly offset, so no money is leftover, and money doesn’t magically appear out of thin air.