 72.1: What is the average of 4.2, 2.61, and 3.6?
 72.2: Four tablespoons equals 14 cup. How many tablespoonswould equal one...
 72.3: The temperature on the moon ranges from a high of about 130Cto a lo...
 72.4: Four of the 12 marbles in the bag are blue. If one marble is taken ...
 72.5: The diameter of a circle is 1 meter. The circumference is how manyc...
 72.6: What fraction of a dollar is a nickel?
 72.7: Find each unknown number:n 12 35
 72.8: Find each unknown number:1  w 712
 72.9: Find each unknown number:w 212 313
 72.10: Find each unknown number:1 w = 0.23
 72.11: Write the standard decimal number for the following:(6 10) a4 110b ...
 72.12: Which of these numbers is closest to 1? A 1 B 0.1 C 10
 72.13: What is the largest prime number that is less than 100?
 72.14: Which of these figures is not a parallelogram? A C B D
 72.15: A loop of string two feet around is formed to make a square. a. How...
 72.16: Figure ABCD is a rectangle. a. Name an angle complementary to DCM. ...
 72.17: Refer to this menu and the information that follows to answerproble...
 72.18: Refer to this menu and the information that follows to answerproble...
 72.19: Refer to this menu and the information that follows to answerproble...
 72.20: If A = lw, and if l equals 2.5 and w equals 0.4, what does A equal?
 72.21: Write the prime factorization of both the numerator and the denomin...
 72.22: Refer to the coordinate plane below to answer problems 22 and 23.Id...
 72.23: Refer to the coordinate plane below to answer problems 22 and 23.Na...
 72.24: Draw a pair of parallel lines. Then draw a second pair of parallell...
 72.25: 12 56 35
 72.26: 3 x 1 12 223
 72.27: 34 2
 72.28: 112 123
 72.29: (0.12)(0.24)
 72.30: 0.6 0.25
Solutions for Chapter 72: Fractions Chart Multiplying Three Fractions
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 72: Fractions Chart Multiplying Three Fractions
Get Full SolutionsSince 30 problems in chapter 72: Fractions Chart Multiplying Three Fractions have been answered, more than 35159 students have viewed full stepbystep solutions from this chapter. Chapter 72: Fractions Chart Multiplying Three Fractions includes 30 full stepbystep solutions. 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. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1.

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Factorization
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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } 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!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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 ().

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.