Madison College Textbook for College Mathematics 804-107 Madison College’s College Mathematics Textbook Page 1 of 256 Table of Contents Table of Contents ................................................................................................................................................................................... 2 Chapter 1: Pre-Algebra ........................................................................................................................................... 3 Section 1.0: Calculator Use .................................................................................................................................................................... 3 Section 1.1: Fractions ............................................................................................................................................................................. 3 Section 1.2: Decimals .......................................................................................................................................................................... 11 Section 1.3: Significant Digits ............................................................................................................................................................. 19 Section 1.4: Signed Numbers ............................................................................................................................................................... 26 Section 1.5: Exponents ......................................................................................................................................................................... 32 Section 1.6: Order of Operations ......................................................................................................................................................... 42 Section 1.7: Evaluating and Simplifying Expressions ......................................................................................................................... 45 Chapter 2: Linear Equations and Inequalities ....................................................................................................... 59 Section 2.1: Solving Linear Equations in One Variable ...................................................................................................................... 59 Section 2.2: Rearranging Formulas and Solving Literal Equations ..................................................................................................... 64 Section 2.3: Linear Inequalities in One Variable ................................................................................................................................. 70 Section 2.4: Applied Problems ............................................................................................................................................................. 71 Section 2.5: Percent Problems ............................................................................................................................................................. 75 Section 2.6: Percent Problems from Finance ....................................................................................................................................... 82 Section 2.7: Direct and Inverse Variation Problems ............................................................................................................................ 85 Chapter 3: Algebra and the Graph of a Line ......................................................................................................... 90 Section 3.1: Graphing a Linear Equation Using a Table of Values ..................................................................................................... 90 Section 3.2: Graphing a Linear Equation Using the Slope Intercept Method ...................................................................................... 93 Section 3.3: Graphing a Linear Equation Using Intercepts .................................................................................................................. 98 Section 3.4: Graphing a Linear Inequality ......................................................................................................................................... 102 Section 3.5: Solving a System of Two Linear Equations by Graphing .............................................................................................. 106 Section 3.6: Solving a System of Two Linear Equations by Algebraic Methods .............................................................................. 113 Chapter 4: Measurement ..................................................................................................................................... 123 Section 4.1: Linear Measurements ..................................................................................................................................................... 123 Section 4.2: Measuring Area .............................................................................................................................................................. 133 Section 4.3: Measuring Volume ......................................................................................................................................................... 138 Section 4.4: Conversion Between Metric and English Units ............................................................................................................. 143 Chapter 5: Geometry ........................................................................................................................................... 150 Section 5.1: Plane geometry............................................................................................................................................................... 150 Section 5.2: Radian Measure and its Applications ............................................................................................................................. 171 Section 5.3: The Volume and Surface Area of a Solid ...................................................................................................................... 174 Chapter 6 - Trigonometry ................................................................................................................................... 184 Section 6.1 Sine, Cosine and Tangent ............................................................................................................................................... 184 Section 6.2 Solving Right Triangles ................................................................................................................................................. 189 Section 6.3 The Law of Sines and the Law of Cosines ..................................................................................................................... 194 Section 6.4 Solving Oblique Triangles ............................................................................................................................................. 200 Chapter 7 - Statistics ........................................................................................................................................... 208 Section 7.1 Organizing Data .............................................................................................................................................................. 208 Section 7.2 Graphing ......................................................................................................................................................................... 218 Section 7.3 Descriptive Statistics ....................................................................................................................................................... 234 Madison College’s College Mathematics Textbook Page 2 of 256 Chapter 1: Pre-Algebra Section 1.0: Calculator Use Throughout most of human history computation has been a tedious task that was often postponed or avoided entirely. It is only in the last generation that the use of inexpensive handheld calculators has transformed the ways that people deal with quantitative data. Today the use and understanding of electronic computation is nearly indispensable for anyone engaged in technical work. There are a variety of inexpensive calculators available for student use. Some even have graphing and/or symbolic capabilities. Most newer model calculators such as the Casio models fx-300W, fx-300MS, fx-115MS, and the Texas Instruments models TI-30X IIB, TI-30X IIS, TI-34 II enter calculations in standard “algebraic” format. Older calculators such as the Casio fx-250HC and the Texas Instruments TI-30Xa and TI-36X enter some calculations in a “reverse” format. Both types of calculators are priced under twenty dollars, yet possess enough computational power to handle the problems faced in most everyday applications. The Casio fx-300MS is fairly representative of the “newer” format calculators and the TI-30Xa is typical of “older” format ones. While other calculator models have similar or even better features for performing the required computations, the reader will be responsible for learning their detailed use. Never throw away the user’s manual! In order to perform a computation, the correct keystrokes must be entered. Although calculators differ in the way keystrokes are entered, this text attempts to provide the reader with a couple of different keystroke options for each example problem in this chapter. The reader should practice the order in which to press the keys on the calculator while reading through the examples. This practice will ensure that the reader knows how to use his/her particular calculator. In order to indicate the sequence of keystrokes the following notation will be used. Digits (0 through 9 plus any decimal point) will be presented in normal typeface. Any additional keystrokes will be enclosed in boxes. For example, to multiply 7 times 8, the command sequence will be written as 7 × 8 = and 56 appears on the display. Section 1.1: Fractions Fractions are ratios of whole numbers, which allow us to express numbers which are between the whole numbers. For example, 2 2 2 =2+ is between 2 and 3 . 3 3 Fractions represent “part of a whole”. Imagine that we have a freight car with eight equal sized compartments. If three of these compartments are full of grain, we would indicate that we have three eighths of a freight car’s worth of grain. This is illustrated below. Consider a car with eight compartments of which two are full. The fraction of a full car is two eighths. If we look at the same car split into four equal compartments, this same amount of grain fills one fourth of the car. We arrive at the following result. Madison College’s College Mathematics Textbook Page 3 of 256 We say that such equal fractions while they “look different” are equivalent. To generate equivalent fractions, we can multiply or divide both numerator (the top number) and denominator (the bottom number) by a common number. So we have the following fractions equivalent to two thirds. 2 2×3 6 = = 3 3×3 9 2×17 34 = = 3×17 51 18 18÷6 3 Similarly, = = .This same result could be stated in terms of “canceling” the 24 24÷6 4 common factor of 6 between the numerator and denominator. 1 8 6×3 6/×3 3 = = = . 2 4 6×4 6/×4 4 If a fraction has no common factors between its numerator and denominator, the fraction is in lowest terms. There are three types of fractions. 1. Proper fractions with the numerator less than (symbolized by < ) the denominator. All proper fractions are less than 1 . 2. Improper fractions with the numerator greater than (symbolized by > ) the denominator. All improper fractions are greater than 1 . Improper fractions can be expressed as a mixed number, which is a whole number plus a proper fraction. For example, 25 1 1 =6 25 =4+ =4 . 6 6 6 3. Unit fractions with the numerator equal to the denominator. All unit fractions are equal to one. For example, 19 4 25 1 = = = =1 . 19 4 25 1 Madison College’s College Mathematics Textbook Page 4 of 256 1 Note: when we write 4 , we are using a shorthand notation. There really is a + sign between the 4 and the one 6 sixth that’s understood but unstated. Working backwards we can convert a mixed number into an improper fraction. For example, 2 7×3+2 23 7 = = . 3 3 3 b Fractions can be entered on many calculators using the a key. For example, use the following keystrokes to c 14 b enter the fraction : 14 a 24 = . 24 c 7 12 will then appear in the display as the fraction reduced to lowest terms. 5 For a mixed number such as, 11 enter the following keystrokes: 6 b b 11 a 5 a 16 = . Some calculators display 11 5 6 , c c while other calculators display 11_5 6 . To change this answer to the improper 181 b b fraction , enter shift a on some calculators , or 2nd a on other calculators. Also, on some 16 c c b b calculators the a key when pressed after entering a fraction converts it to a decimal and if a is pressed a c c second time the decimal is converted back to a fraction. On other calculators fraction – decimal conversions are performed by entering 2nd ← . The ← key is also the back space key, which deletes characters in the display. 6 6 As an application, solve for the following missing numerator: = . As a first step reduce to lowest 12 8 8 3 3 terms as . So = . The first denominator 12 is three times the second denominator 4, so the missing 4 12 4 numerator must be three times the second numerator 3 . The answer is that the missing numerator is 9. To compare two fractions and determine which is larger, we can use the following procedure: 1. If the fractions involve mixed numbers with proper fractions, the number with the larger whole number is the larger number. For example, 3 7 7 >5 , since 7>5 . 16 8 2. If the fractions are both proper fractions or mixed numbers with equal whole numbers, then convert the fractions into decimals. The number with the larger decimal is the larger number. For example, Madison College’s College Mathematics Textbook Page 5 of 256 1 3 3 1 5 <5 , since =0.375>0.333...= . 3 8 8 3 3 1 To add or subtract fractions we need a common denominator. Consider adding to . Since one fourth is equivalent to two eighths, we have the follo8wing 4solution: If mixed numbers are involved, we first deal with the whole numbers, then the fractions. For example, 1 9 1 9 8 9 16 8 9 16+8−9 15 7 −5 =7−5+ − =2+ − =1+ + − =1+ =1 . 2 16 2 16 16 16 16 16 16 16 16 9 8 16 Note: Since > , we had to “borrow” from the 2. 16 16 16 To multiply fractions we form the product of the numerators over the product of the denominators. For example, 5 3 5×3 15 × = = . 8 4 8×4 32 If the product involves mixed numbers, we first convert them to improper fractions. For example, 5 1 2×6+5 4×5+1 17 21 17 3/×7 17×7 119 9 2 ×4 = × = × = × = = =11 . 6 5 6 5 6 5 3/×2 5 2×5 10 10 Note: we canceled the common factor of 3 between numerator and denominator in this calculation. In a multiplication problem this can always be done and saves the effort of later having to reduce the final answer. Also note that the answer is “reasonable” in that 5 1 5 1 2 ≈3 and 4 ≈4 , so 2 ×4 ≈3×4=12 . 6 5 6 5 A quick estimation like this can often catch silly mistakes even when using a calculator! 8 8 1 1 Consider the division problem8÷2=4 .This is the same as = × =8× =4 . 2 1 2 2 More generally, any division problem can be expressed as Madison College’s College Mathematics Textbook Page 6 of 256 a a 1 1 a÷b= = × =a× . b 1 b b This means that division by the number b is equivalent to multiplication by the fraction 1 1 b . The fraction is called the reciprocal of b= . To form the reciprocal of a number we exchange the b b 1 numerator with the denominator. In summary, division by a non-zero number equals multiplication by the reciprocal of that number. In a division problem 0 is never allowed as the denominator or divisor. The reason for this is as follows. 20 Suppose20÷0= made sense. Then there would be some number, a, which is the 0 answer to this division problem. Restating this as a multiplication problem would give a×0=20 . But any 20 number times zero gives zero! So no sensible answer to 20÷0= exists. 0 20 Another way of explaining this goes to the very meaning of division. 20÷4= =5 , 4 means that 20 contains five 4’s. How many 0’s does 20 contain? There’s no sensible answer to the question! 1 1 Consider now the division ÷ , from the diagram below it is clear that one fourth 4 8 contains 2 one eighths. So the answer must be 2 . The following shows that this result is consistent with the multiplication by the reciprocal definition of division. 1 1 1 8 8 ÷ = × = =2 . 4 8 4 1 4 If the division involves mixed numbers, we first convert them into improper fractions. For example, 1 1 9 9 9/ 8 8 4 ÷1 = ÷ = × = =4 . 2 8 2 8 2 9/ 2 Madison College’s College Mathematics Textbook Page 7 of 256 In expressions which combine operations the standard order of operations apply as shown in the following: 2 1 1 2/ 5 5 1 5 5 5×4 5 20−5 15 3 3/×1 1 ×2 −1 ÷3= × − × = − = − = = =1 =1 =1 . 3 2 4 3 2/ 4 3 3 12 3×4 12 12 12 12 3/×4 4 These calculations are all easily performed on the calculator. The keystokes for the previous calculation are as follows: b b b b b 2 a 3 × 2 a 1 a 2 − 1 a 1 a 4 ÷ 3 = . c c c c c More involved calculations with grouping symbols are also possible. For example, 3  3 7 1 3  3 14 5 3  16+3−14 2 3 −5 −3  ÷2 =3 −2+ − ÷ =3 −1+ × = 4  16 8 2 4  16 16 2 4  16  5 3 5 2 3 21 2 3 21 30-21 =3 −1 × =3 − × =3 − = 3+ 4 16 5 4 16 5 4 40 40 9 =3 . 40 This is keystroked as follows: b b b b b b b b 3 a 3 a 4 − ( 5 a 3 a 16 − 3 a 7 a 8 ) ÷ 2 a 1 a 2 = . c c c c c c c c Your Turn!! Write as an improper fraction. 11 6 1) ________________ 16 7 3 2) ________________ 10 Write as a mixed number reduced to lowest terms. 19 3) ________________ 8 4) 26 ________________ 10 Madison College’s College Mathematics Textbook Page 8 of 256 Reduce to lowest terms. 9 5) ________________ 12 18 7 6) ________________ 32 Supply the missing numerators: 3 ? 7) ________________ = 4 12 3 ? 3 = 8) ________________ 4 8 Indicate which number is larger. 19 5 32 9) ________________ 8 7 3 3 2 10) ________________ 16 4 Perform the indicated operations and express the answer as a fraction in lowest terms: 3 4 × 11) ________________ 16 9 2 6÷ 12) ________________ 3 3 5 + 13) ________________ 8 8 13 9 − 14) ________________ 32 32 Madison College’s College Mathematics Textbook Page 9 of 256 1 1 4 ×2 15) ________________ 8 4 5 15 ÷ 16) ________________ 8 16 3 11 5 +3 17) ________________ 8 32 3 3 5 ÷2 18) ________________ 8 4 1 2 2 ×3 19) ________________ 4 3 19 3 5 7 −5 −1 20) ________________ 32 4 8 1 1 5 3 ×3 −10 21) ________________ 2 4 8 3 3 1 −4 ÷4 22) ________________ 8 4 Solve and state all results as fractions reduced to lowest terms. 5 How many pieces of inch thick plywood are in a stack 35 inches high? 16 23) ________________ A lumberyard sells lumber only in even foot lengths. What is the shortest single board of 1 3 lumber from which a carpenter could cut three 3 ft long and two 2 ft long pieces? 4 4 24) ________________ 1 A cubic foot contains about 7 gallons. How many cubic feet are there in 120 gallons? 2 25) ________________ 1 3 A nail3 inches long, goes through a board2 inches thick supporting a joist. How far into the joist does the 2 8 nail extend? Madison College’s College Mathematics Textbook Page 10 of 256