 15.3.15.1.91: The total population under consideration divided by the number of i...
 15.3.15.1.92: When each groups population is divided by the standard divisor, a s...
 15.3.15.1.93: A standard quota rounded down to the nearest whole number is called...
 15.3.15.1.94: A standard quota rounded up to the nearest whole number is called a...
 15.3.15.1.95: The rule stating that an apportionment should always be either the ...
 15.3.15.1.96: The apportionment method that requires rounding the standard quota ...
 15.3.15.1.97: Jeffersons method, Websters method, and Adamss method require using...
 15.3.15.1.98: a) The apportionment method that uses a modified divisor that is le...
 15.3.15.1.99: a) The apportionment method that uses a modified quota that is alwa...
 15.3.15.1.100: Jeffersons method, Websters method, and Adamss method all make use ...
 15.3.15.1.101: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.102: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.103: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.104: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.105: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.106: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.107: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.108: Legislative Seats In Exercises 1118, suppose that Turtlestan is a s...
 15.3.15.1.109: Hotel Staff In Exercises 1926, a large hotel chain needs to apporti...
 15.3.15.1.110: Hotel Staff In Exercises 1926, a large hotel chain needs to apporti...
 15.3.15.1.111: Hotel Staff In Exercises 1926, a large hotel chain needs to apporti...
 15.3.15.1.112: Hotel Staff In Exercises 1926, a large hotel chain needs to apporti...
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 15.3.15.1.115: Hotel Staff In Exercises 1926, a large hotel chain needs to apporti...
 15.3.15.1.116: Hotel Staff In Exercises 1926, a large hotel chain needs to apporti...
 15.3.15.1.117: Umbrellas In Exercises 2730, Sandy Shores Resorts operates four bea...
 15.3.15.1.118: Umbrellas In Exercises 2730, Sandy Shores Resorts operates four bea...
 15.3.15.1.119: Umbrellas In Exercises 2730, Sandy Shores Resorts operates four bea...
 15.3.15.1.120: Umbrellas In Exercises 2730, Sandy Shores Resorts operates four bea...
 15.3.15.1.121: New Computers In Exercises 3134, a university is made up of five sc...
 15.3.15.1.122: New Computers In Exercises 3134, a university is made up of five sc...
 15.3.15.1.123: New Computers In Exercises 3134, a university is made up of five sc...
 15.3.15.1.124: New Computers In Exercises 3134, a university is made up of five sc...
 15.3.15.1.125: New Cars In Exercises 3538, a car manufacturer has 150 cars of a ne...
 15.3.15.1.126: New Cars In Exercises 3538, a car manufacturer has 150 cars of a ne...
 15.3.15.1.127: New Cars In Exercises 3538, a car manufacturer has 150 cars of a ne...
 15.3.15.1.128: New Cars In Exercises 3538, a car manufacturer has 150 cars of a ne...
 15.3.15.1.129: New Buses In Exercises 39 42, the Transit Department in the city of...
 15.3.15.1.130: New Buses In Exercises 39 42, the Transit Department in the city of...
 15.3.15.1.131: New Buses In Exercises 39 42, the Transit Department in the city of...
 15.3.15.1.132: New Buses In Exercises 39 42, the Transit Department in the city of...
 15.3.15.1.133: Nursing Shifts In Exercises 43 46, a hospital has 200 nurses to be ...
 15.3.15.1.134: Nursing Shifts In Exercises 43 46, a hospital has 200 nurses to be ...
 15.3.15.1.135: Nursing Shifts In Exercises 43 46, a hospital has 200 nurses to be ...
 15.3.15.1.136: Nursing Shifts In Exercises 43 46, a hospital has 200 nurses to be ...
 15.3.15.1.137: The First Census The first census taken in the United States after ...
 15.3.15.1.138: Legislative Seats Suppose that a country with a population of 10,00...
 15.3.15.1.139: Police Officers Suppose that a police department has 210 new office...
 15.3.15.1.140: Do research and write a report on the apportionment method used in ...
 15.3.15.1.141: Do research and write a report on the HuntingtonHill method, the cu...
Solutions for Chapter 15.3: Voting and Apportionment
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 15.3: Voting and Apportionment
Get Full SolutionsChapter 15.3: Voting and Apportionment includes 51 full stepbystep solutions. Since 51 problems in chapter 15.3: Voting and Apportionment have been answered, more than 80376 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.