Mom found an open box of her children's favorite candy bars. She decides to apportion the candy bars among her three youngest children according to the number of minutes each child spend doing homework during the week. (a) Suppose that there were 10 candy bars in the box. Given that Bob did homework for a total of 54 minutes, Peter did homework for a total of 243 minutes, and Ron did homework for a total of 703 minutes, apportion the 10 candy bars among the children using Hamilton's method. (b) Suppose that just before she hands out the candy bars, mom finds one extra candy bar. Using the same total minutes as in (a), apportion now the 11 candy bars among the children using Hamilton's method. (c) Two results of (a) and (b) illustrate one of the paradoxes of Hamilton's method. Which one? Explain.

How many ways are there to distribute 36 pieces of chocolate equally among 4 children?

How does the idea of a "light at the center of every cell" tie in with the speaker's thoughts?

Find an equation in standard form for the line described. Having y-intercept 6 and parallel to the x-axis

Prove that for any function f on [a, b],

$$R _ { N } - L _ { N } = \frac { b - a } { N } ( f ( b ) - f ( a ) )$$

Anne, Bette, and Chia jointly own a flower shop. They can't get along anymore and decide to break up the partnership using the method of sealed bids, with the understanding that one of them will get the flower shop and the other two will get cahs. Anne bids $210,000, Bette bids$240,000, and Chia bids $225,000. How much money do Anne and Chia each get from Bette for their third share of the flower shop?

a. A country has two states, state A, with a population of 9450, and state B, with a population of 90,550. The congress has 100 seats, divided between the two states according to their respective populations. Use Hamilton’s method to apportion the congressional seats to the states. b. Suppose that a third state, state C, with a population of 10,400, is added to the country. The country adds 10 new congressional seats for state C. Use Hamilton’s method to show that the new-states paradox occurs when the congressional seats are reapportioned.

Three partners {A, B, and C) jointly own a business but wish to dissolve the partnership using the method of sealed, bids. One of the partners will keep the business; the other two will each get cash for their one-third share of the business. Suppose that A bids $x, B bids$y, and C bids $z. Assume that C is the high bidder. Describe the final settlement of this fair division in terms of x, y, and z.

This exercise comes in two parts. Read Part I and answer (a) and (b), then read Part II and answer (c) and (d). Part 1. A catering company contracts to provide catering services to three schools: Alexdale, with 617 students BromviUe, with 1,292 students, and Canley, with 981 students. The 30 food-service workers employed by the catering company are apportioned among the schools based on student enrollments. (a) Find the standard divisor,rounded lo the nearest integer. (b) Find the apportionment of the 30 workers to the three schools under Hamilton's method. Part 11. The catering company gets a contract to service one additional school-Dillwood, with 885 students To account for the additional students, the company hires 9 additional food-service workers. [885 students represent approximately 9 workers based on the standard divisor found m(a).] (c) Find the apportionment of the 39 workers to the four schools under Hamilton's method. (d) Which paradox is illustrated by the results of (b) and (c) ? Explain.

Matter is anything that has mass and takes up space.

If the standard quota of state X is 35.41, then which of the following apportionments to state X is (or are) possible under Jefferson's method? (a) 34, 35, or 36, (b) 34, 35, 36, or 37, (c) 35 only, (d) 36 only, (e) 35, 36, or 37.

$$\langle \mathbf { u } , \mathbf { v } \rangle$$

defines an inner product on

$$\mathbb R^2,$$

where

$$\mathbf { u } = \left[ \begin{array} { l } { u _ { 1 } } \\ { u _ { 2 } } \end{array} \right] \text { and } \mathbf { v } = \left[ \begin{array} { l } { v _ { 1 } } \\ { v _ { 2 } } \end{array} \right].$$

Find a symmetric matrix A such that

$$\langle \mathbf { u } , \mathbf { v } \rangle = \mathbf { u } ^ { T } A \mathbf { v }.$$

$$\langle \mathbf { u } , \mathbf { v } \rangle = 4 u _ { 1 } v _ { 1 } + u _ { 1 } v _ { 2 } + u _ { 2 } v _ { 1 } + 4 u _ { 2 } v _ { 2 }$$

Four partners (Adams, Benson, Gagle, and Duncan) jointly own a piece of land with a market value of $400,000. Suppose that the land is subdivided into four parcels$ $s_1, s_2, s_3$ $, and$ $s_4$ $. The partners are planning to split up, with each partner getting one of the four parcels. (a) To Adams,$ $s_1$ $is worth$40,00 more than

$$s_2,s_2$$

and

$$s_3$$

are equal in value, and

$$s_4$$

is worth $20.000 more than$ $s_1$ $. Determine which of the four parcels are fair shares to Adams. (b0 To Benson,$ $s_1$ $is worth$40,000 more then

$$s_2, s_4$$

is $8,000 more than$ $s_3$ $, and together$ $s_4$ $and$ $s_3$ $have a combined value equal to 40$ $s_1$ $is worth$40,000 more than

$$s_2$$

and $20,000 more than$ $s_4$ $, and$ $s_3$ $is worth twice as much as$ $s_4$ $. Determine which of the four parcels are fair shares to Cagle. (d) To Duncan,$ $s_1$ $is worth$4,000 more than

$$s_2, s_2$$

and

$$s_3$$

have equal value; and

$$s_1, s_2$$

and

$$s_3$$

have a combined value equal to 70% of the value of the land. Determine which of the four parcels are fair shares to Duncan. (e) Find a fair division of the land using the parcels

$$s_1, s_2, s_3$$

, and

$$s_4$$

as fair shares.

Explain how to calculate the standard divisor of an apportionment for a total population p with n items to apportion.