 22.1: When the sum of 15 and 12 is subtracted from the product of 15 and1...
 22.2: There were 13 original states. There are now 50 states. What fracti...
 22.3: A marathon race is 26 miles plus 385 yards long. A mile is 1760 yar...
 22.4: If 23 of the 12 apples were eaten, how many were eaten? Draw adiagr...
 22.5: What number is 34 of 16? Draw a diagram to illustrate theproblem.
 22.6: How much money is 310 of $3.50? Draw a diagram to illustratethe pro...
 22.7: As Shannon rode her bike out of the low desert, the elevation chang...
 22.8: Find each unknown number. Check your work.w 15 = 8
 22.9: Find each unknown number. Check your work.w15 345
 22.10: Find each unknown number. Check your work.36 + $4.78 + $34.09
 22.11: Find each unknown number. Check your work.$12.45 3
 22.12: Find each unknown number. Check your work.35 1000
 22.13: Find each unknown number. Check your work.7 9 1435 1000 3
 22.14: Shannon bought three dozen party favors for $1.24 each. Toestimate ...
 22.15: Which digit in 375,426,198,000 is in the tenmillions place?
 22.16: Find the greatest common factor of 12 and 15.
 22.17: List the whole numbers that are factors of 30.
 22.18: The number 100 is divisible by which of these numbers: 2, 3, 5, 9, 10?
 22.19: Jeb answered 45 of the questions correctly. What percent of theques...
 22.20: Compare: 13 12
 22.21: Which of these numbers is not a prime number? A 19 B 29 C 39
 22.22: (3 + 3) (3 3)
 22.23: Find the number halfway between 27 and 43.
 22.24: What is the perimeter of the rectangle below?
 22.25: Use an inch ruler to find the length of the line segment below.
 22.26: Corn bread and wheat bread were baked in pans of equalsize. The cor...
 22.27: Compare these fractions. Draw and shade rectangles to illustrate th...
 22.28: A quarter of a year is 14 of a year. There are 12 months in ayear. ...
 22.29: A bit is one eighth of a dollar. a. How many bits are in a dollar? ...
 22.30: he letters c, p, and t represent three different numbers.When p is ...
Solutions for Chapter 22: Equal Groups Problems with Fractions
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 22: Equal Groups Problems with Fractions
Get Full SolutionsChapter 22: Equal Groups Problems with Fractions includes 30 full stepbystep solutions. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. Since 30 problems in chapter 22: Equal Groups Problems with Fractions have been answered, more than 35733 students have viewed full stepbystep solutions from this chapter. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose 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).

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 picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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