- 6-2.1: 48% of 12 is what number?
- 6-2.2: 32% of what number is 28?
- 6-2.3: What percent of 158 is 47.4?
- 6-2.4: What number is 130% of 149?
- 6-2.5: 15% of what number is 80?
- 6-2.6: 48% of what number is 120?
- 6-2.7: Find Pif and
- 6-2.8: Find 40% of 160.
- 6-2.9: What number is of 150?
- 6-2.10: What number is 154% of 30?
- 6-2.11: Find Bif and
- 6-2.12: Find Rif and B = 280.
- 6-2.13: 40% of 30 is what number?
- 6-2.14: 52% of 17.8 is what number?
- 6-2.15: 30% of what number is 21?
- 6-2.16: 17.5% of what number is 18? Round to hundredths.
- 6-2.17: What percent of 16 is 4?
- 6-2.18: What percent of 50 is 30?
- 6-2.19: 172% of 50 is what number?
- 6-2.20: 0.8% of 50 is what number?
- 6-2.21: What percent of 15.2 is 12.7? Round to the nearest hundredth of a p...
- 6-2.22: What percent of 73 is 120? Round to the nearest hundredth of a perc...
- 6-2.23: 0.28% of what number is 12? Round to the nearest hundredth.
- 6-2.24: 1.5% of what number is 20? Round to the nearest hundredth.
- 6-2.25: At the Evans Formal Wear department store, all suits are reduced 20...
- 6-2.26: Joe Passarelli earns $8.67 per hour working for Dracken Internation...
- 6-2.27: An ice cream truck began its daily route with 95 gallons of ice cre...
- 6-2.28: Stacy Bauer sold 80% of the tie-dyed T-shirts she took to the Green...
- 6-2.29: A stockholder sold her shares and made a profit of $1,466. If this ...
- 6-2.30: The Drammelonnie Department Store sold 30% of its shirts in stock. ...
- 6-2.31: Ali gave correct answers to 23 of the 25 questions on the driving t...
- 6-2.32: A soccer stadium in Manchester, England, has a capacity of 78,753 s...
- 6-2.33: Holly Hobbs purchased a magazine at the Atlanta airport for $2.99. ...
- 6-2.34: A receipt from Wal-Mart in Memphis showed $4.69 tax on a subtotal o...
Solutions for Chapter 6-2: SOLVING PERCENTAGE PROBLEMS
Full solutions for Business Math, | 9th Edition
Remove row i and column j; multiply the determinant by (-I)i + j •
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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!).
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.