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# class+February+25 - Math 103 Section 11 Friday Chapter 3...

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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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