$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 18.4: Fair Division

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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

18.4: Fair Division is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.