1. KU122: Introduction to Math Skills and Strategies Prof. Scott Brown Unit 4 Seminar Developmental Mathematics Bittinger, Beecher 4

2. Announcements/Reminders You MUST show EACH of the steps in your project to get full credit. You can either show it in words or using numbers/symbols.

3. Unit 4 Assignments The following assignments are due by 11:59pm ET on Tuesday: 1.) Live seminar 2.) Discussion 3.) Practice Problems 4.) MML Ungraded Tutorials 5.) MML Quiz

4. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Percent Notation 4.1 Ratio and Proportion 4.2 Percent Notation 4.3 Percent and Fraction Notation 4.4 Solving Percent Problems Using Percent Equations 4.5 Solving Percent Problems Using Proportions 4.6 Applications of Percent 4.7 Sales Tax, Commission, Discount, and Interest 4.8 Interest Rates on Credit Cards and Loans 4 CHAPTER

5. Slide 4- 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Find fraction notation for ratios.

6. Slide 4- 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ratio A ratio is the quotient of two quantities. For every 50 chicks raised, 3 die within the first two days. The ratio of the number of chicks that die to the number raised is shown by fraction notation

7. Slide 4- 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A Find the ratio of 5 to 6. Solution The ratio is

8. Slide 4- 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example B Find the ratio of 25.6 to 100. Solution The ratio is

9. Slide 4- 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example C For every 5 people having a dog for a pet 2 others have a cat. What is the ratio of dog owners to cat owners? Solution The ratio is

10. Slide 4- 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example D Refer to the triangle. a) What is the ratio of the length of the shortest side to the length of the longest side? b) What is the ratio of the length of the longest side to the length of the shortest side? Solution a) b) 5 ft 12 ft 13 ft

11. Slide 4- 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective B Give the ratio of two different measures as a rate.

12. Slide 4- 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A 2005 Kia Sportage EX 4WD can go 414 miles on 18 gallons of gasoline. Let’s consider the ratio of miles to gallons: = 23 miles per gallon = 23 mpg. The ratio is called a rate. “per” means division, or “for each.”

13. Slide 4- 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rate When a ratio is used to compare two different kinds of measure, we call it a rate.

14. Slide 4- 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example E It takes 1 quart (32 ounces) of fertilizer to cover 6400 square feet of tall fescue grass. What is the rate in ounces per square foot? Solution

15. Slide 4- 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective C Determine whether two pairs of numbers are proportional.

16. Slide 4- 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When two pairs of numbers such as (3, 2 and 6, 4) have the same ratio, we say that they are proportional. The equation states that the pairs 3, 2 and 6, 4 are proportional. Such an equation is called a proportion. We call 3 · 4 and 2 · 6 cross products.

17. Slide 4- 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example G Determine whether 2, 4, and 5, 10 are proportional. Solution We can use cross products to check an equivalent equation: 2  10 5  4 2  10 = 5  4 20 = 20 Since the last equation is true, we know that the first equation is also true. The numbers 2, 4 and 5, 10 are proportional.

18. Slide 4- 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example H Determine whether 6, 7 and 8, 9 are proportional. Solution We can use cross products to check an equivalent equation: 6  9 8  7 6  9 = 8  7 54  56 Since 54  56, we know 6, 7 and 8, 9 are not proportional.

19. Slide 4- 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective D Solve proportions.

20. Slide 4- 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Proportions To solve equate cross products and then divide on both sides to get x alone.

21. Slide 4- 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example J Solve: Write a mixed numeral for the answer. Solution Equating cross products Dividing both sides by 5 Check: and

22. Slide 4- 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example K Solve: Write decimal notation for the answer. Solution Equating cross products Dividing by 3.6 Simplifying Dividing

23. Slide 4- 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective E Solve applied problems involving proportions.

24. Slide 4- 24 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example L If 5 ounces of a medicine must be mixed with 8 ounces of water, how many ounces of medicine would be mixed with 36 ounces of water? Solution Let m represent how much medicine would be needed. Thus, 22.5 ounces of medicine would be needed.

25. Slide 4- 25 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example M On a road atlas, 1 in. represents 22.5 miles. If two cities are 4.5 in. apart on the map, how far apart are they in reality? Solution 1. Familiarize. We let r = the distance apart 2. Translate.

26. Slide 4- 26 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example M 3. Solve. 4. Check. We substitute into the proportion and check cross products. The cross products are the same. 5. State. The actual distance apart is 101.25 miles.

27. Slide 4- 27 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example N To determine the number of deer in a wildlife preserve, a conservationist catches 275 deer, tags them, and releases them back into the wild. Later, 120 deer are caught, and 21 of them are found to be tagged. Estimate how many deer are in the wildlife preserve. Familiarize. Our strategy is to form two different ratios with can be used to represent the ratio of tagged deer to all the deer. One ratio is Number of tagged deer/Number of deer in preserve. A second way is to assume that the deer are uniformly distributed in the preserve. That ratio could be Number of tagged deer caught/Total number of deer caught.

28. Slide 4- 28 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example N Translate. We translate the proportion as follows: Solve.

29. Slide 4- 29 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example N Check. We substitute into the proportion and check cross products: State. We estimate that there are 1571 deer in the wildlife preserve. The cross products are very close.

30. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PERCENT NOTATION A - Write three kinds of notation for a percent. B - Convert between percent notation and decimal notation. 4.2 4.2

31. Slide 4- 31 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Write three kinds of notation for a percent.

32. Slide 4- 32 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 40 of 100 squares are shaded. 40% or or 0.40 of the large square is shaded.

33. Slide 4- 33 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Percent notation is used extensively in our everyday lives. Here are some examples: 63% of the aluminum used in the United States is recycled. 0.08% blood alcohol level is a standard used by most states at the legal limit for drunk driving. 33% of all U.S. citizens say the day they dread most is the day they go to the dentist.

34. Slide 4- 34 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Percent notation is often represented in pie charts to show how the parts of a quantity are related. To draw the pie chart think of a pie cut into 100 equally sized pieces. Then shade a wedge equal in size to 56 of the pieces to represent 56%. Shade in a wedge equal to 14 of the pieces to represent 14%, and so on.

35. Slide 4- 35 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Percent Notation The notation n% means “n per hundred.”

36. Slide 4- 36 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Notation for n % Percent notation, n%, can be expressed using: ratio  n% = the ratio of n to 100 = fraction notation  decimal notation 

37. Slide 4- 37 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A Write three kinds of notation for 45%. Solution Using ratio: Using fraction notation: Using decimal notation:

38. Slide 4- 38 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective B Convert between percent notation and decimal notation.

39. Slide 4- 39 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To write decimal notation for a number like 23%, we can replace the “%” with “  0.01” and multiply: 23% = 23  0.01 = 0.23 Similarly, 5.8% = 5.8  0.01 = 0.058 324% = 324  0.01 = 3.24

40. Slide 4- 40 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To convert from percent notation to decimal notation, a) replace the percent symbol with  0.01, and b) multiply by 0.01, which means move the decimal point two places to the left. 36.5% 36.5  0.01 Move 2 places to the left 0.36.5 36.5% = 0.365

41. Slide 4- 41 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example C Write an equivalent decimal for 67.34%. Solution a) Replace the % symbol with  0.01. 67.34  0.01 b) Multiply to move the decimal point two places to the left.0.67.34 Thus, 67.34% = 0.6734.

42. Slide 4- 42 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example D Write an equivalent decimal for 2.3%. Solution a) Replace the % symbol with  0.01. 2.3  0.01 b) Multiply to move the decimal point two places to the left. Thus, 2.3% = 0.023. 0.02.3 This zero serves as a placeholder.

43. Slide 4- 43 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To convert from decimal notation to percent notation, multiply by 100%. That is, a) move the decimal point two places to the right, and b) write a % symbol. 0.675 = 0.675  100% Move 2 places to the right 0.67.5 67.5% 0.675 = 67.5%

44. Slide 4- 44 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example E Write percent notation for 0.6. Solution a) Multiply by 100 to move the decimal point two places to the right. 0.60. b) Write a % symbol. 60% Thus 0.6 = 60% This zero serves as a place holder.

45. Slide 4- 45 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example F Write percent notation for 2.35 Solution a) Multiply by 100 to move the decimal point two places to the right. 2.35. b) Write a % symbol. 235% Thus 2.35 = 235%

46. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PERCENT AND FRACTION NOTATION A - Convert from fraction notation to percent notation. B - Convert from percent notation to fraction notation. 4.3 4.3

47. Slide 4- 47 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Convert from fraction notation to percent notation.

48. Slide 4- 48 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To convert from fraction notation to percent notation, a) find decimal notation by division, and b) convert the decimal notation to percent notation. 0.6 = 0.60 = 60% Percent notation

49. Slide 4- 49 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A Find percent notation for Solution a) Find decimal notation by division. b) Convert the decimal notation to percent notation. To do so, multiply by 100 to move the decimal point two places to the right, and write a % symbol.

50. Slide 4- 50 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example B Write as an equivalent percent. Solution We use the definition of percent as a ratio.

51. Slide 4- 51 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective B Convert from percent notation to fraction notation.

52. Slide 4- 52 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To convert from percent notation to fraction notation, a) use the definition of percent as a ratio, and b) simplify, if possible. 30% Percent notation

53. Slide 4- 53 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example C Find percent notation for 34% and simplify. Solution Using the definition of percent Simplify by removing a factor equal to 1.

54. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SOLVING PERCENT PROBLEMS USING PERCENT EQUATIONS A - Translate percent problems to percent equations. B - Solve basic percent problems. 4.4 4.4

55. Slide 4- 55 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Translate percent problems to percent equations.

56. Slide 4- 56 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To solve a problem involving percents, it is helpful to translate first to an equation. For example, “23% of 5 is what?” 23% of 5 is what?      0.23  5 = a Note how the key words are translated.

57. Slide 4- 57 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Key Words in Percent Translations “Of” translates to “  ” or “  ”. “Is” translates to “=”. “What” translates to any letter. “%” translates to

58. Slide 4- 58 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A Translate: What is 19% of 82? Solution What is 19% of 82?      a = 19%  82

59. Slide 4- 59 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example B Translate: 7 is 10% of what? Solution 7 is 10% of what?      7 = 10%  b

60. Slide 4- 60 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example C Translate: 18 is what percent of 38? Solution 18 is what percent of 38      18 = p  38

61. Slide 4- 61 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective B Solve basic percent problems.

62. Slide 4- 62 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley In solving percent problems, we use the Translate and Solve steps in the problem-solving strategy used throughout the text. Amount = Percent number × Base.

63. Slide 4- 63 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Types of Percent Problems 1. Finding the amount (the result of taking the percent) Example: What is 25% of 60? Translation: a = 25%  60 2. Finding the base (the number you are taking the percent of) Example: 15 is 25% of what number? Translation: 15 = 25%  b 3. Finding the percent number (the percent itself) Example: 15 is what percent of 60? Translation: 15 = p  60

64. Slide 4- 64 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example D: Finding the Amount What is 8% of 34? Solution Translate: a = 8% × 34 Solve: The variable is by itself. To solve the equation, we just convert 8% to decimal notation and multiply. a = 0.08 × 34 a = 2.72 Thus, 2.72 is 8% of 34. The answer is 2.72.

65. Slide 4- 65 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example E: Finding the Base 15 is 16% of what? Solution Translate: 15 is 16% of what? 15 = 16% × b Solve: To solve we divide both sides of the equation by 16%:

66. Slide 4- 66 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example F: Finding the Percent 27 is what percent of 36? Solution Translate: 27 is what percent of 36? 27 = p × 36 Solve: To solve we divide both sides by 36 and convert the answer to percent notation:

67. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SOLVING PERCENT PROBLEMS USING PROPORTIONS A - Translate percent problems to proportions. B - Solve basic percent problems. 4.5 4.5

68. Slide 4- 68 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Translate percent problems to proportions.

69. Slide 4- 69 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A percent is a ratio of some number to 100. For example, 75% is the ratio The numbers 3 and 4 have the same ratio as 75 and 100. Thus, To solve a percent problem using a proportion, we translate as follows: Number  100   Amount  Base You might find it helpful to read this a “part is to whole as part is to whole.” 75% 3 4 75 100

70. Slide 4- 70 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A Translate to a proportion. 15% of 7 is what? Solution 15% of 7 is what?

71. Slide 4- 71 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example B Translate to a proportion. What is 120% of 73? Solution What is 120% of 73?

72. Slide 4- 72 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example C Translate to a proportion. 18 is 12% of what? Solution 18 is 12% of what?

73. Slide 4- 73 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example D Translate to a proportion. 35% of what is 21? Solution 35% of what is 21?

74. Slide 4- 74 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example E Translate to a proportion. 22 is what percent of 50? Solution 22 is what percent of 50?

75. Slide 4- 75 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example F Translate to a proportion. What percent of 72 is 9? Solution What percent of 72 is 9?

76. Slide 4- 76 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective B Solve basic percent problems.

77. Slide 4- 77 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example G Solve: 35% of what is 21? Solution35% of what is 21? Translate Solve 35 35% of 60 is 21

78. Slide 4- 78 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example H Solve: 15% of 7 is what? Solution15% of 7 is what? Translate Solve 100 15% of 7 is 1.05

79. Slide 4- 79 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example I Solve: 22 is what percent of 50? Solution 22 is what percent of 50? Translate Solve 50 22 is 44% of 50

80. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley APPLICATIONS OF PERCENT A - Solve applied problems involving percent. B - Solve applied problems involving percent of increase or decrease. 4.6 4.6

81. Slide 4- 81 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Solve applied problems involving percent.

82. Slide 4- 82 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A There have been a total of 54 presidential elections. In some cases a president was re-elected to office accounting for our 43 presidents. The president’s who died in office were elected in 13 elections of the 54. In what percent of the elections did the president die? Familiarize. The question asks for the percent of presidents who died. Estimate 13 is about 1/5 of 54. Translate. 13 is what percent of 54? 13 = p  54

83. Slide 4- 83 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued example A Solve. 13 = p  54 54 54 Check. Our answer is close to our estimate of 1/5 or 20%. State. About 24% of the U.S. Presidents died in all 54 elections. or

84. Slide 4- 84 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective B Solve applied problems involving percent of increase or decrease.

85. Slide 4- 85 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What does it mean to say that the price of oil has decreased 8%? If the price was $7.00 and it went down to $6.44, then the decrease is $0.56, which is 8% of the original price. $7.00 $6.44 100% 92% 8% $0.56

86. Slide 4- 86 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example B A family pays a monthly electric bill of $172.00. With careful monitoring they can reduce their bill to $163.40. What is the percent of decrease? Familiarize. We find the amount of decrease and then make a drawing. $172.00 $163.40 100% ?%  $8.60

87. Slide 4- 87 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example B Translate. We rephrase and translate. 8.60 is what percent of 172.00? 8.60 = p  172.00 Solve. We divide both sides by 172:

88. Slide 4- 88 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example B Check. To check, we note that, with a 5% decrease, the reduced bill should be 95% of the original bill. Since 95% of 172 = 0.95(172) = 163.40, our answer checks. State. The percent of decrease of the electric bill is 5%.

89. Slide 4- 89 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example C A computer technician earns a starting salary of $46,000 for one year and receives a 4% raise the following year. What is the new salary? Familiarize. We note that the amount of the raise can be found and then added to the starting salary. A drawing can help us visualize this. We let x = the new salary. $46,000 100% 4% $?

90. Slide 4- 90 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example C Translate. We rephrase the question and translate as follows. What is the starting salary plus 4% of the starting salary? x = 46,000 + 0.04  46,000 Solve. x = 46,000 +.04(46,000) = 46,000 + 1840 = 47,840

91. Slide 4- 91 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example C Check. To check, we note that the new salary is 100% of the starting salary plus 4% of the starting salary. Thus the new salary is 104% of the starting salary. Since 1.04(46,000) = 47,840, our answer checks. State. The new salary is $47,840.

92. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SALES TAX, COMMISSION, DISCOUNT, AND INTEREST 4.7 4.7 A - Solve applied problems involving sales tax and percent. B - Solve applied problems involving commission and percent. C - Solve applied problems involving discount and percent. D - Solve applied problems involving simple interest. E - Solve applied problems involving compound interest.

93. Slide 4- 93 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Solve applied problems involving sales tax and percent.

94. Slide 4- 94 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sales tax computations represent a special type of percent of increase problem. Sales Tax Sales Tax = Sales tax rate × Purchase price Total price = Purchase price + sales tax

95. Slide 4- 95 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A The sales tax rate in Ohio is 6%. How much tax is charged on the purchase of 4 books at $15.95 each? What is the total price? Solution a) We first find the cost of the books. 4 × $15.95 = $63.80 b) The sales tax on items costing $63.80 is Sales tax rate × Purchase price 0.06 × $63.80 or $3.83

96. Slide 4- 96 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example A The total price is given by the purchase price plus the sales tax: $63.80 + $3.83, or $67.63 Check: The total price is 106% of the purchase price. Since 63.80 × 1.06  67.63, we have a check. The sales tax is $3.83 and the total price is $67.63

97. Slide 4- 97 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example B The sales tax is $140 on the purchase of a new swimming pool which cost $1750. What is the sales tax rate? Solution Rephrase: Sales tax is what percent of purchase price? Translate: $140 = p × $1750 To solve, divide both sides by 1750 The sales tax rate is 8%.

98. Slide 4- 98 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example C The sales tax on a digital camera is $32.18 and the sales tax rate is 7.5%. Find the purchase price (the price before taxes are added). Solution Rephrase: Sales tax is 7.5% of what? Translate: 32.18 = 0.075 × c To solve, we divide both sides by 0.075: The purchase price is $429.07.

99. Slide 4- 99 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective B Solve applied problems involving commission and percent.

100. Slide 4- 100 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When you work for a salary, you receive the same amount of money each week or month. When you work for commission, you are paid a percentage of the total sales you complete. Commission Commission = Commission rate × Sales

101. Slide 4- 101 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example D A person’s sales commission is 8%. What is the commission from the sale of $73,230 worth of new car sales? Solution Commission = Commission rate × Sales C = 8% × 73,230 C = 0.08 × 73,230 C = 5858.40 The commission is $5858.40.

102. Slide 4- 102 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example E Rebecca earns a commission of $17,340 from selling a $289,000 home. What is the commission rate? Solution Commission = Commission rate × Sales 17,340 = r × 289,000 To solve, we divide both sides by 289,000: The commission rate is 6%.

103. Slide 4- 103 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example F Sam’s commission is 9%. She receives a commission of $1260 on the sale of pharmaceuticals. How much did the pharmaceuticals cost? Solution Commission = Commission rate × Sales 1260 = 0.09 × S To solve: divide both sides by 0.09: Sam sold $14,000 worth of pharmaceuticals.

104. Slide 4- 104 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective C Solve applied problems involving discount and percent.

105. Slide 4- 105 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Suppose that the regular price of a rug is $60, and the rug is on sale at 25% off. Since 25% of 60 is $15, the sale price is $60 – $15, or $45. We call $60 the original, or marked price, 25% the rate of discount, $15 the discount, and $45 the sale price. Discount and Sale Price Discount = Rate of discount × Original price Sale Price = Original price  Discount

106. Slide 4- 106 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example G A couch marked $875 is on sale at 25% off. What is the discount? What is the sale price? Solution a) Discount = Rate of discount × Original price D = 25% × 875 D = 0.25 × 875 D = 218.75 b) Sale price = Original price – Discount = 875 – 218.75 = 656.25 The discount is $218.75 and the sale price is $656.25.

107. Slide 4- 107 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example H A dome tent is marked down from $329 to $213.85. What is the rate of discount? Solution Find the discount by subtracting the sale price from the original price: 329 – 213.85 = 115.15 The discount is 115.15.

108. Slide 4- 108 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued example H Discount = Rate of discount × Original price 115.15 = r × 329 To solve, we divide both sides by 329. The discount rate is 35%.

109. Slide 4- 109 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective D Solve applied problems involving simple interest.

110. Slide 4- 110 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Suppose you put $1000 into an investment for 1 year. The $1000 is called the principal. If the interest rate is 8%, in addition to the principal, you get back 8% of the principal which is 8% of $1000, or 0.08 × $1000, or $80.00. The $80.00 is called the interest, or more precisely, the simple interest. It is, in effect the price that a financial institution pays for the use of the money over time. Simple Interest Formula The simple interest I on the principal P, invested for t years at interest rate r, is given by I = P  r  t,

111. Slide 4- 111 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example I What is the simple interest on $4500 invested at an interest rate of 7% for 1 year? Solution We use the formula I = P  r  t: I = P  r  t = $4500 × 0.07 × 1 = $4500 × 0.07 = $315 The simple interest for 1 year is $315.

112. Slide 4- 112 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example J What is the simple interest on a principal of $4500 invested at an interest rate of 7% for ¼ year? Solution We use the formula I = P  r  t: I = P  r  t = $4500 × 7% × The simple interest for ¼ year is $78.75.

113. Slide 4- 113 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example K To pay for a shipment of coffee mugs, Café Mocha borrows $4000 at 7 ½% for 75 days. Find (a) the amount of simple interest that is due and (b) the total amount that must be paid after 75 days. Solution We express 75 days as a fractional part of a year:

114. Slide 4- 114 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example K The interest due for 75 days is $61.64. b) The total amount to be paid after 75 days is the principal plus the interest: $4000 + $61.64 = $4061.64. The total amount due is $4061.64.

115. Slide 4- 115 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective E Solve applied problems involving compound interest.

116. Slide 4- 116 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When interest is paid on interest, we call it compound interest. This is the type of interest usually paid on investments or loans. Suppose you have $5000 in a savings account at 6%. In 1 year, the account will contain the original $5000 plus 6% of $5000. Thus, the total in the account after 1 year will be 106% of $5000, or 1.06 × $5000, or $5300. Now, suppose that the total of $5300 remains in the account for another year. At the end of the second year, the account will contain $5300 plus 6% of $5300. 106% of $5300, or 1.06 × $5300, or $5618.

117. Slide 4- 117 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Note that in the second year, interest is earned on the first year’s interest. When this happens, we say that interest is compounded annually.

118. Slide 4- 118 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example L Find the amount in an account if $3000 is invested at 5%, compounded annually for 2 years. Solution a) After 1 year, the account will contain 105% of $3000: 1.05 × $3000 = $3150 b) At the end of the second year, the account will contain 105% of $3150: 1.05 × $3150 = $3307.50 The amount in the account after 2 years is $3307.50.

119. Slide 4- 119 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Compound Interest Formula If the principal P has been invested at interest rate r, compounded n times a year, in t years it will grow to an amount A given by

120. Slide 4- 120 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example M The Murray’s invest $9000 in an account paying 6% compounded quarterly. Find the amount in the account after 3 ½ years. Solution The compounding is quarterly, so n = 4. P = $9000 r = 0.06 t = 3 ½

121. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley INTEREST RATES ON CREDIT CARDS AND LOANS 4.8 4.8 A - Solve applied problems involving interest rates on credit cards and loans.

122. Slide 4- 122 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective A Solve applied problems involving interest rates on credit cards and loans.

123. Slide 4- 123 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Comparing interest rates is essential if one is to become financially responsible. A small change in an interest rate can make a large difference in the cost of a loan.

124. Slide 4- 124 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A After the holidays, Rose has a balance of $3724.72 on a credit card with an annual percentage rate (APR) of 21.1%. She decides not to make any additional purchases with this card until she has paid off the balance. a) Many credit cards require a minimum monthly payment of 2% of the balance. What is Rose’s minimum monthly payment on a balance of $3724.72? b) For the minimum payment found in part (a), find the amount of interest and the amount applied to reduce the principal. c) If Sara had the same balance on a credit card with an APR of 12.2%, how much of her first payment would be interest? How much would be applied to reduce the principal? d) Compare the amounts for 12.2% with the amounts for 21.1%.

125. Slide 4- 125 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example A a) Many credit cards require a minimum monthly payment of 2% of the balance. What is Rose’s minimum monthly payment on a balance of $3724.72? Solution Multiply the balance by 2% 0.02 × $3724.72 = $74.49 Rounded to the nearest dollar is $74.

126. Slide 4- 126 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example A b) For the minimum payment found in part (a), find the amount of interest and the amount applied to reduce the principal. Solution The amount of interest on $3724.72 at 21.1% for one month: We subtract to find the amount applied to reduce the principal in the first payment: Amt. applied = Minimum payment – Interest = $74 – $65.49 = $8.51

127. Slide 4- 127 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example A c) If Sara had the same balance on a credit card with an APR of 12.2%, how much of her first payment would be interest? How much would be applied to reduce the principal? Solution Subtract to find amount applied to principal. Amt. applied = Minimum payment – Interest = $74 – $37.87 = $36.13

128. Slide 4- 128 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example A d) Compare the amounts for 12.2% with the amounts for 21.1%. BalanceFirst Mo. Pay % APRAmt. of Interest Amt. applied to Prin. Balance after 1 st pay $3724.72$7421.1%$65.49$8.51$3716.21 3724.727412.237.8736.133688.59

129. Slide 4- 129 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example B The Martin’s recently purchased their first home. They borrowed $125,000 at 7 ½% for 30 years (360 payments). Their monthly payment (excluding insurance and taxes) is $875. a) How much of their first payment is interest and how much is applied to reduce the principal? b) If the Martin’s pay the entire 360 payments, how much interest will be paid on the loan?

130. Slide 4- 130 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example B a) How much of their first payment is interest and how much is applied to reduce the principal? The portion of their first payment applied to the principal is: $875 – $781.25 = $93.75.

131. Slide 4- 131 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Example B b) If the Martin’s pay the entire 360 payments, how much interest will be paid on the loan? Over the 30-year period, the total paid will be 360 × $875 = $315,000. The total amount of interest paid over the lifetime of the loan is $315,000 – $125,000 = $190,000.