- 105.1: How far would a car travel in 212 hours at 50 miles per hour?50 mi1...
- 105.2: A map of Texas is drawn to a scale of 1 inch = 50 miles.Houston and...
- 105.3: The ratio of humpback whales to orcas was 2 to 1. If there were900 ...
- 105.4: When Robert measured a half-gallon box offrozen yogurt, he found it...
- 105.5: Calculate mentally: a. +10 + 10 b. 10 10 c. +6 + 5 4
- 105.6: On Earth a 1-kilogram object weighs about 2.2 pounds.A rock weighs ...
- 105.7: Sonia has only dimes and nickels in her coin jar; they are in a rat...
- 105.8: The airline sold 25% of the seats on the plane at a discount.If 45 ...
- 105.9: Complete the table to answer problems 911.350 a. b.
- 105.10: Complete the table to answer problems 911.a. 0.04 b.
- 105.11: Complete the table to answer problems 911.a. b. 150%
- 105.12: 4 112 516 214
- 105.13: 45 313 4 3
- 105.14: 0.125 80
- 105.15: (1 + 0.5) (1 0.5)
- 105.16: Solve: c12 34
- 105.17: What is the total cost of an $8.75 purchase after 8% sales taxis ad...
- 105.18: Write the decimal number one hundred five and five hundredths.
- 105.19: The measure of A in quadrilateralABCD is 115. What are the measures...
- 105.20: Write the prime factorization of 500 using exponents.
- 105.21: A quart is a little less than a liter, so a gallon is a little less...
- 105.22: Diane will spin the spinner twice. What is theprobability that it w...
- 105.23: The perimeter of this isosceles triangle is18 cm. What is the lengt...
- 105.24: What is the area of the triangle in problem 23?
- 105.25: The temperature was 5F at 6:00 a.m. By noon the temperature hadrise...
- 105.26: The weather report stated that the chance of rain is 30%. Use a dec...
- 105.27: Find the perimeter of this figure. Dimensionsare in inches.
- 105.28: Study this function table anddescribe the rule that helps you find ...
- 105.29: A room is 15 feet long and 12 feet wide. a. The room is how many ya...
- 105.30: Ned rolled a die and it turned up 6. If he rolls the die again,what...
Solutions for Chapter 105: Using Proportions to Solve Percent Problems
Full solutions for Saxon Math, Course 1 | 1st Edition
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).
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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 .
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.
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).