- 6-3.1: A number increased from 5,286 to 7,595. Find the amount of increase.
- 6-3.2: A number decreased from 486 to 104. Find the amount of decrease.
- 6-3.3: Find the amount of increase if 432 is increased by 25%.
- 6-3.4: Find the amount of decrease if 68 is decreased by 15%.
- 6-3.5: If 135 is decreased by 75%, what is the new amount?
- 6-3.6: If 78 is increased by 40%, what is the new amount?
- 6-3.7: A number increased from 224 to 336. Find the percent of increase.
- 6-3.8: A number decreased from 250 to 195. Find the rate of decrease.
- 6-3.9: A number is decreased by 40% to 525. What is the original amount?
- 6-3.10: A number is increased by 15% to 43.7. Find the original amount.
- 6-3.11: The cost of a pound of nails increased from $2.36 to $2.53. What is...
- 6-3.12: Wrigley announced the first increase in 16 years in the price of a ...
- 6-3.13: Bret Davis is getting a 4.5% raise. His current salary is $38,950. ...
- 6-3.14: Kewanna Johns plans to lose 12% of her weight in the next 12 weeks....
- 6-3.15: DeMarco Jones makes $13.95 per hour but is getting a 5.5% increase....
- 6-3.16: Carol Wynne bought a silver tray that originally cost $195 and was ...
- 6-3.17: A laptop computer that was originally priced at $2,400 now sells fo...
- 6-3.18: Federated Department Stores dropped the price of a winter coat by 1...
Solutions for Chapter 6-3: INCREASES AND DECREASES
Full solutions for Business Math, | 9th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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.)
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
lA-II = 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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.