 88.1: Mavis scored 12 of the teams 20 points. What percent of the teamspo...
 88.2: One fourth of an inch of snow fell every hour during the storm. How...
 88.3: Eamon wants to buy a new baseball glove that costs $50. He has $14a...
 88.4: Find the area of this triangle.
 88.5: Draw a ratio box for this problem. Then solve the problemusing a pr...
 88.6: What is the volume of this rectangularprism?
 88.7: Find each unknown factor:10w = 25
 88.8: Find each unknown factor:20 = 9m
 88.9: a. What is the perimeter of this triangle? b. What is the area of t...
 88.10: Write 5% as a a. decimal number. b. fraction.
 88.11: Write 25 as a decimal, and multiply it by 2.5. What is the product?
 88.12: Compare:23 3223 32
 88.13: 13 1001
 88.14: 6 112
 88.15: 12 0.25
 88.16: 0.025 100
 88.17: If the tax rate is 7%, what is the tax on a $24.90 purchase?
 88.18: The prime factorization of what number is 22 32 52?
 88.19: Which of these is a composite number? A 61 B 71 C 81 D 101
 88.20: Round the decimal number one and twentythree hundredths to thenear...
 88.21: Albert baked 5 dozen muffins and gave away712 of them. How manymuff...
 88.22: 6 3 6 3
 88.23: How many milliliters is 4 liters?
 88.24: Draw a line segment 2 14 inches long. Label the endpoints A andC. T...
 88.25: On a coordinate plane draw a rectangle with vertices at (2, 2), (4,...
 88.26: What is the ratio of the length to the width of the rectangle inpro...
 88.27: How do you calculate the area of a triangle?
 88.28: In the figure at right, angles ADB and BDCare supplementary.a. What...
 88.29: Nathan drew a circle with a radiusof 10 cm. Then he drew a square a...
 88.30: Write a word problem that can be solved using the proportion68 w100...
Solutions for Chapter 88: Using Proportions to Solve Ratio Word Problems
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 88: Using Proportions to Solve Ratio Word Problems
Get Full SolutionsThis 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 88: Using Proportions to Solve Ratio Word Problems have been answered, more than 36065 students have viewed full stepbystep solutions from this chapter. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. Chapter 88: Using Proportions to Solve Ratio Word Problems includes 30 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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 column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.