18.4: Fair Division
1. Chance values the veggie half at $7.50 and pepperoni half at $2.50.
A full pepperoni slice is \(\frac{1}{4}\) of the pepperoni half. Value \(\$ 2.50 / 4=\$ 0.625\)
A full veggie slice is \(\frac{1}{4}\) of the veggie half. Value \(\$ 7.50 / 4=\$ 1.875\)
A slice that is ½ pepperoni \(\frac{1}{2}\) veggie is value \(\$ 0.3125+\$ 0.9375=\$ 1.25\)
3. Erin: Bowl 1, Catherine: Bowl 2, Shannon: Bowl 3
5. a. 25 Snickers @ $0.01 each, 20 Milky Ways @ $0.05 each, 60 Reese’s @ $0.02 each
Value: \(\$ 0.25+\$ 1.00+\$ 1.20=\$ 2.45\)
b. No. Dustin values the whole bag at $8, so a fair share would be $4.
c. Lots of possibilities. Here’s a couple:
80 Milky Ways, 0 Snickers, 0 Reese’s
50 Snickers, 50 Milky Ways, 50 Reese’s
7. a. Zoe
b. Maggie: s2, s3. Meredith: s1, s2. Holly: s3
c. Maggie: s2, Meredith: s1, Holly: s3, Zoe: s4
9. a. P5
b. $6.50 (doesn’t need to trim it much since they’re last)
c. P4 would receive it, with value $6.00 (since P4 would trim it)
11. a. \((320+220) / 4=\$ 135\)
b. Desk and Vanity both go to A. A pays \(\$ 320+\$ 220-\$ 135=\$ 405\) to estate
B gets $95, C gets $125, D gets $110.
c. Surplus of \(\$ 405-\$ 95-\$ 125-\$ 110=\$ 75\) gets split, $18.75 each.
A gets desk and vanity, pays $386.25 to estate
B gets $113.75, C gets $143.75, D gets $128.75
13. Fair shares: Abby: 10.333, Ben: 9, Carla: 7.667
Motorcycle to Abby, Car to Ben, Tractor to Abby, Boat to Abby
Initial: Abby pays $10.667, Ben pays $2, Carla gets $7.667
Surplus: $5; $1.667 each
Final: Abby gets Motorcycle, Tractor and Boat, pays $9
Ben gets Car, pays $0.333
Carla gets $9.334
15. Fair shares: Sasha: $135, Megan: $140
Sasha gets: Couch, detail cleaning. Value $80
Megan gets: TV, Stereo, carpets. Value: $260
Initial: Sasha gets $55, Megan pays $120.
Surplus: $65; $32.50 each
Final: Sasha gets Couch and does detail cleaning, gets $87.50
Megan gets TV and stereo, and cleans carpets, pays $87.50
17. a. s3, worth $270
b. s1 and s4 have combined value $440 for Greedy, so piece would be worth $220