 Chapter 1: REVIEW OF WHOLE NUMBERS AND INTEGERS
 Chapter 11: PLACE VALUE AND OUR NUMBER SYSTEM
 Chapter 12: OPERATIONS WITH WHOLE NUMBERS AND INTEGERS
 Chapter 10: PAYROLL
 Chapter 101: GROSS PAY
 Chapter 102: PAYROLL DEDUCTIONS
 Chapter 103: THE EMPLOYERS PAYROLL TAXES
 Chapter 11: SIMPLE INTEREST AND SIMPLE DISCOUNT
 Chapter 111: THE SIMPLE INTEREST FORMULA
 Chapter 112: ORDINARY AND EXACT INTEREST
 Chapter 113: PROMISSORY NOTES
 Chapter 12: CONSUMER CREDIT
 Chapter 121: INSTALLMENT LOANS AND CLOSEDEND CREDIT
 Chapter 122: PAYING A LOAN BEFORE IT IS DUE: THE RULE OF 78
 Chapter 123: OPENEND CREDIT
 Chapter 13: COMPOUND INTEREST, FUTURE VALUE, AND PRESENT VALUE
 Chapter 131: COMPOUND INTEREST AND FUTURE VALUE
 Chapter 132: PRESENT VALUE
 Chapter 14: ANNUITIES AND SINKING FUNDS
 Chapter 141: FUTURE VALUE OF AN ANNUITY
 Chapter 142: SINKING FUNDS AND THE PRESENT VALUE OF AN ANNUITY
 Chapter 15: BUILDING WEALTH THROUGH INVESTMENTS
 Chapter 151: STOCKS
 Chapter 152: BONDS
 Chapter 153: MUTUAL FUNDS
 Chapter 16: MORTGAGES
 Chapter 161: MORTGAGE PAYMENTS
 Chapter 162: AMORTIZATION SCHEDULES AND QUALIFYING RATIOS
 Chapter 17: DEPRECIATION
 Chapter 171: DEPRECIATION METHODS FOR FINANCIAL STATEMENT REPORTING
 Chapter 172: DEPRECIATION METHODS FOR IRS REPORTING
 Chapter 18: INVENTORY
 Chapter 181: INVENTORY
 Chapter 182: TURNOVER AND OVERHEAD
 Chapter 19: INSURANCE
 Chapter 191: LIFE INSURANCE
 Chapter 192: PROPERTY INSURANCE
 Chapter 193: MOTOR VEHICLE INSURANCE
 Chapter 2: REVIEW OF FRACTIONS
 Chapter 21: FRACTIONS
 Chapter 22: ADDING AND SUBTRACTING FRACTIONS
 Chapter 23: MULTIPLYING AND DIVIDING FRACTIONS
 Chapter 20: TAXES
 Chapter 201: SALES TAX AND EXCISE TAX
 Chapter 202: PROPERTY TAX
 Chapter 203: INCOME TAXES
 Chapter 21: FINANCIAL STATEMENTS
 Chapter 211: THE BALANCE SHEET
 Chapter 212: INCOME STATEMENTS
 Chapter 213: FINANCIAL STATEMENT RATIOS
 Chapter 3: DECIMALS
 Chapter 31: DECIMALS AND THE PLACEVALUE SYSTEM
 Chapter 32: OPERATIONS WITH DECIMALS
 Chapter 33: DECIMAL AND FRACTION CONVERSIONS
 Chapter 4: BANKING
 Chapter 41: CHECKING ACCOUNT TRANSACTIONS
 Chapter 42: BANK STATEMENTS
 Chapter 5: EQUATIONS
 Chapter 51: EQUATIONS
 Chapter 52: USING EQUATIONS TO SOLVE PROBLEMS
 Chapter 53: FORMULAS
 Chapter 6: PERCENTS
 Chapter 61: PERCENT EQUIVALENTS
 Chapter 62: SOLVING PERCENTAGE PROBLEMS
 Chapter 63: INCREASES AND DECREASES
 Chapter 7: BUSINESS STATISTICS
 Chapter 71: GRAPHS AND CHARTS
 Chapter 72: MEASURES OF CENTRAL TENDENCY
 Chapter 73: MEASURES OF DISPERSION
 Chapter 8: TRADE AND CASH DISCOUNTS
 Chapter 81: SINGLE TRADE DISCOUNTS
 Chapter 82: TRADE DISCOUNT SERIES
 Chapter 83: CASH DISCOUNTS AND SALES TERMS
 Chapter 9: MARKUP AND MARKDOWN
 Chapter 91: MARKUP BASED ON COST
 Chapter 92: MARKUP BASED ON SELLING PRICE AND MARKUP COMPARISONS
 Chapter 93: MARKDOWN, SERIES OF MARKDOWNS, AND PERISHABLES
Business Math, 9th Edition  Solutions by Chapter
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Business Math,  9th Edition  Solutions by Chapter
Get Full SolutionsThis textbook survival guide was created for the textbook: Business Math, , edition: 9. The full stepbystep solution to problem in Business Math, were answered by , our top Math solution expert on 03/08/18, 08:36PM. Business Math, was written by and is associated to the ISBN: 9780135108178. Since problems from 77 chapters in Business Math, have been answered, more than 15489 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 77.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.