 25.1: What is the difference between the sum of 12 and 12 and the sumof 1...
 25.2: In three tries Carlos punted the football 35 yards, 30 yards, and37...
 25.3: Earths average distance from the Sun is one hundred fortynine mill...
 25.4: What is the perimeter of the rectangle?
 25.5: A 30inch length of ribbon was cut into 4 equal lengths. Howlong wa...
 25.6: Two thirds of the class finished the test on time. What fraction of...
 25.7: Compare: 12 of 12 13 of 12
 25.8: What fraction is half of the fraction that is half of 12?
 25.9: A whole circle is 100% of a circle. What percent of a circle is 19 ...
 25.10: How many 16 s are in 1? How many 16 s are in 12?
 25.11: What fraction of a circle is 33 13% of a circle?
 25.12: Divide 365 by 7 and write the answer as a mixed number.
 25.13: 23 23
 25.14: 66 56
 25.15: 30 40 60
 25.16: 512 512
 25.17: A team won seven of the twenty games played and lost the rest. What...
 25.18: Cheryl bought 10 pens for 25 each. How much did she payfor all 10 p...
 25.19: What is the greatest common factor (GCF) of 24 and 30?
 25.20: What number is 1100 of 100?
 25.21: Find each unknown number. Check your work.58 m 1
 25.22: Find each unknown number. Check your work.144n 12
 25.23: What is the sum of 3142, 6328, and 4743 to the nearestthousand?
 25.24: Two thirds of the 60 students liked peaches. How manyof the student...
 25.25: Estimate the length in inches of the line segment below. Thenuse an...
 25.26: To divide a circle into thirds, Jan imagined the circle was theface...
 25.27: Write154 as a mixed number.
 25.28: Draw and shade rectangles to illustrate and complete thiscomparison...
 25.29: What are the first four multiples of 25?
 25.30: Which of these numbers is divisible by both 9 and 10?How do you kno...
Solutions for Chapter 25: Writing Division Answers as Mixed Numbers Multiples
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 25: Writing Division Answers as Mixed Numbers Multiples
Get Full SolutionsChapter 25: Writing Division Answers as Mixed Numbers Multiples includes 30 full stepbystep solutions. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 30 problems in chapter 25: Writing Division Answers as Mixed Numbers Multiples have been answered, more than 35396 students have viewed full stepbystep solutions from this chapter. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.