Math 103, Section 11, Friday, February 25, 2011
Chapter 3 Handout 2
The Mathematics of Sharing
Assignment Chapter 3A
Please do Exercises 12,18,30, and 32.
Homework will be submitted to Sakai.
Homework to be discussed in class on Friday February 25.
Exercise 7,9, and 11
Exercise #10 on page 105.
Three players (Alex, Betty, and Cindy) are sharing a cake. Suppose that
the cake is divided into three slices (s1,s2, and s3) . The following table
shows the value of s1 and of s2 to each of the players. The values of s3
are missing. (The percentages represent the value of the slice as a percent
of the value of the entire cake.)
s1
s2
s3
Fair share
value
Fair
shares
Alex
30%
34%
Betty
28%
36%
Cindy
30%
33 1/3 %
Which of the three slices are fair shares to Alex?
a.
Which of the three slices are fair shares to Betty?
b.
Which of the three slices are fair shares to Cindy?
c.
If possible, find a fair division of the cake using s11, s2, and s3 as fair
shares. If this is not possible, explain why not.
Solution:
s1
s2
s3
Fair share
value
Fair
shares
Alex
30%
34%
36%
33
1/3 %
Betty
28%
36%
36%
33
1/3 %
Cindy
30%
33 1/3 %
36 2/3%
33
1/3 %
Find a fair division, if possible:
7.
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S1
S2
S3
Fair share value
Fair
shares
Ana
$3.00
$5.00
$4.00
(3+5+4)/3=$4.
S2,S3
Ben
$4.00
$4.50
$6.50
(4+4.50+6.50)/3=
$5
S3
Cara
$4.50
$4.50
$4.50
$4.50
S1,S2,S3
Describe the fair division of the cake.
9.
S1
S2
S3
Fair share
value
Fair
shares
Alex
30%
40%
30%
33 1/3 %
S2
Betty
31%
35%
34%
33 1/3 %
S2,S3
Cindy
30%
35%
35%
33 1/3 %
S2,S3
Is there a fair division?
11.
S1
S2
S3
S4
Fair
share
value
Fair
shares
Abe
$3.00
$5.00
$5.00
$2.00
$3.75
S2,S3
Betty
$4.50
$4.50
$4.50
$4.50
$4.50
S1,S2,S3,
S4
Cory
$4.00
$3.50
$2.00
$2.50
$3.00
S1,S2
Dana
$2.75
$2.40
$2.45
$2.40
$2.50
S1
Is there a fair division? How many fair divisions are there?
Fair division problems can be classified three ways depending on the nature of the booty S.
•
Continuous.
The set S can be divided infinitely many ways. Examples are cakes, pizza,
and land.
•
Discrete.
The set comprises indivisible objects (or objects that are not easily divisible).
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 Spring '07
 Berkowitz
 Math, Inch, fair share, Fair division, a. Jared, b. Karla

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