7th Grade. Percents.

Transcription

1 1

2 7th Grade Percents

3 Table of Contents Click on the topic to go to that section Relating Fractions, Decimals and Percents Three Types of Percent Problems Percent of Change Representing Percent Equations Algebraically Applied Percent of Decrease Teacher Notes Applied Percent of Increase Real life Application Problems Glossary 3

4 Relating Fractions, Decimals & Percents Return to table of contents 4

5 Helping you remember... Fill in each box below with an example of the process described. % to a fraction % to a decimal fraction to a % decimal to a % 5

6 Ordering Order the numbers from least to greatest % In order to do this, they must all be in the same form. Let's turn them all into percents: 15% 12.5% 16% 9.5% So least to greatest: 9.5% 12.5% 15% 16% %

7 1 Find the lowest value. A 5% B 1/2 C.5% D.05 7

8 2 Find the greatest value. A 120% B 1.02 C.2% D

9 3 Find the greatest value. A 6% B.6 C 60 D 6 9

10 4 Find the lowest value. A 2% B.2 C.02 D.2% 10

11 5 Find the lowest value. A 50% B 500% C 50.0 D

12 Examples Express each decimal or percent as a fraction in lowest terms: 1) 18% 2) ) click click click 4) 5) ) click click click 12

13 Examples Express each fraction as a percent: 1) 2) 3) click click click 13

14 6 Express as a fraction. 14

15 7 Express as a decimal. 15

16 8 Express as a percent. 16

17 9 Express as a decimal. 17

18 10 Express as a percent. 18

19 11 Express as a percent. 19

20 Three Types of Percent Problems Return to table of contents 20

21 Types of Percent Problems Remember, percents are "parts of a whole". The part is the numerator and the whole is the denominator. 17% means 17 parts per 100 or We are going to solve problems involving percents. There are 3 types of problems: 1. Find the part What number is 54% of 34? 2. Find the whole 4 is 60% of what number? 3. Find the percent 18 is what percent of 28? 21

22 Two words that will occur in these types of problems are: "is" "of" Types of Percent Problems These words have specific meanings in math. "Is" means equals (=) "Of" means multiply To solve a percent problem, translate the words into an equation. Change the following: 1. Percent into a decimal 2. "is" to "=" 3. "of" to " " 4. Unknown to "x" Then, solve the equation. 22

23 Finding the Part... 23

24 Find the Part Examples: Find 40% of 60 Write a mathematical sentence = 24 Click 20% of 90 Write a mathematical sentence = 18 Click 24

25 Write a Mathematical Sentence What is 10% of 88? X = Try these: X = 8.8 Find 12% of 70 What is 40% of 28? 25

26 Set Up a Proportion Another Method: You can also solve percent problems by setting up a proportion. Since percents are parts of a whole, you can create the following proportion: 26

27 Set Up a Proportion When figuring out which is the "part" and which is the "whole", remember that you take a percent of the whole and the answer is the part. In other words, the whole is with the word "of" and the part is with the word "is". 27

28 Proportion Method Steps 1. Set up the proportion as shown. is of = % 100 Note: You can use this box to solve many problems involving percents! 2. Substitute given values into the proportion. Note: Try to find the numbers that are attached to the words/symbols: is, of, or percent. 3. Solve the proportion. 28

29 Proportion Method Example Example: What is 25% of 400? Steps 1. Set up the proportion. is of? 400 = % Substitute. What is 25% of 400? Click on each box to see if you substituted correctly. 3. Solve. click 400 x 25 = 100w 10,000 = 100w 10,000/100 = w 100 = w 29

30 Proportion Method Example Example: What is 32% of 300? Steps 1. Set up the proportion. is of? 300 = % Substitute. What is 32% of 300? Click on each box to see if you substituted correctly. 3. Solve. 300 x 32 = 100w 9,600 = 100w 9600/100 = w 96 = w click 30

31 Proportion Method Try It Try it: What is 20% of 180? Steps 1. Set up the proportion. is of = % Substitute. 3. Solve. 31

32 12 Find 30% of

33 13 What is 15% of 90? 33

34 14 Find the greater value. A 20% of 16 B 10% of 90 C 25% of 40 D 100% of 7 34

35 15 Find the greater value. A 2% of 1000 B 5% of 500 C 10% of 300 D 15% of

36 16 Identify any values that are equal. A What is 40% of 80? B 60% of 70 C 25% of 128 D 200% of 16 36

37 Finding the Whole... 37

38 Remember, you can solve this by: 1. Translating into an equation 2. Setting up a proportion 40% of what number is 50?.40 X = 50 Finding the Whole X = X =

39 Try This: 100 is 20% of what number? 100 =.20 x 100 = x.20 x = 500 Finding the Whole 39

40 17 56 is 70% of what? 40

41 18 12% of what number is 6? 41

42 19 65% of what number is 10? 42

43 20 27 is 150% of what number? 43

44 21 1% of what number is 12? 44

45 Finding the Percent... 45

46 Finding the Percent Remember, you can solve this by: 1. Translating into an equation 2. Setting up a proportion What percent of 80 is 24? x 80 = 24 X = X =.30 X = 30% 46

47 60 is what percent of 15? 60 = x = X 15 4 = X 400% = X Finding the Percent 47

48 22 What percent of 3 is 12? 48

49 23 30 is what percent of 36? 49

50 24 What percent of 18 is 180? 50

51 25 2 is what percent of 1? 51

52 26 What percent of 25 is 20? 52

53 Percent Problems You have just studied three different types of percent problems. Try all 3 types: 24 is 40% of what number? 42 is what percent of 840? What is 30% of 45? 53

54 27 Find the largest value. A What is 50% of 50? B What number is 45% of 60? C 30 is 60% of what number? D 25% of what number is 150? 54

55 28 Find the greatest percentage value. A What percent of 30 is 18? B 60 is what percent of 90? C What percent of 70 is 210? D 1,000 is what percent of 100? 55

56 29 Find 20% of

57 30 Eight is what percent of 28? 57

58 31 What number is 3% of 17? 58

59 32 Find 27% of

60 33 23 is what percent of 200? 60

61 34 What percent is 35 of 20? 61

62 35 Fifty six percent of what number is 40? 62

63 36 Forty five is 30% of what number? 63

64 37 Sixty two percent of 40 is what number? A 24.8 B.0155 C 24.8% D

65 Percent of Change Return to table of contents 65

66 Percent of Change Percent of Change is the ratio of the amount of increase or decrease to the original amount. It is an increase when the new amount is larger than the original and a decrease when the new amount is smaller than the original. To find the percent of change, use the following proportion: Percent of change: Amount of increase or decrease = % Original Amount

67 Percent of Change Find the percent of change (be sure to label your answer as an increase or decrease). Examples: Original amount: 20 Original amount: 40 New amount: 30 New amount: 10 Percent of change= Percent of change= 67

68 Percent of Change Identify the percent of change as an increase or decrease. Then find the percent of change. 1. Original: 45 New: Original: 100 New: Original: 58 New: 75 68

69 Percent of Change Try This! A CD's original price was $ It is now on sale for $ What is the percent of change? 69

70 Percent of Change Try This! A student's first test grade was 60. The second test grade was an 85. What was the percent of change? 70

71 38 In 2005, the price of a McDonald's hamburger was $0.89. In 2010, the price of a McDonald's hamburger was $1.19. What was the percent of change? 71

72 39 Original Amount: 500 New: 700 Find the percent of change. 72

73 40 Original Amount: 52 New: 17 Find the percent of change. 73

74 41 The number of students who attended FHS in 2010 was In 2011, 1380 students attended FHS. What was the percent of change in student enrollment? 74

75 42 Find the percent of change. Original price: $120 Sale price: $75 75

76 43 Find the percent of change. Original price: $80 Sale price: $50 76

77 44 A stereo, originally priced at $360, is on sale for $200. What is the percent of change? 77

78 Representing Percent Equations Algebraically Return to table of contents 78

79 Representing Percent Equations Algebraically You have already begun translating percent problems into equations. Remember... To solve a percent problem, translate the words into an equation. Change: 1. Percent into a decimal 2. "is" to "=" 3. "of" to " " 4. Unknown to "x" Then, solve the equation. 79

80 Think about this % + 5% = 105% What does that equation look like in decimal form? = 1.05 So, if you increase the price of a shirt 5%, the new price is 105% of the original price. To represent that algebraically, you would write it this way: Let s = the original price of the shirt 1s s = 1.05s 80

81 Representing Percent Equations Algebraically Example: You sell a shirt for $ This price represents a 5% increase from the price you paid for the shirt. How much did it cost you to purchase the shirt? Let s = the original price of the shirt 1s s = s = s = $14.76 The shirt cost you $

82 Representing Percent Equations Algebraically Example: The population of your school decreased by 13% from last year to this year. If there are 957 students in the school this year, how many were there last year? 2 students solved this differently. Who is correct? Why? Is one method easier than the other? Student 1: Student 2: 100% 13% = 87% 1n.13n = % of what is 957? 0.87n = n = 957 n = 1,100 students n = 1,100 students 82

83 Representing Percent Equations Algebraically So, what does this mean? m m = 1.15m This could mean increase m by 15% or multiply m by They mean the same thing! Click Likewise, what is the meaning of w 0.42w = 0.58w This means both decrease w by 42% or multiply w by Click 83

84 Representing Percent Equations Algebraically You Try. 1. A smart phone is on sale for $299, or 18% off. What was the original price of the phone? Write and solve an equation to represent this situation. 2. What does this equation mean? p p = 1.02p 3. What does this equation mean? h 0.1h = 0.9h 84

85 45 Write an equation to represent the problem, then solve. Be prepared to show your equation! When you go shopping, you must pay an additional 6% in sales tax. What is the price of your items before taxes if your final price is $25? 85

86 46 Choose the equation that represents the situation. The population of a town increased by 1%. A x x = 1.01x B x + 0.1x = 1.1x C x 0.1x = 0.9x D x 0.01x = 0.99x 86

87 47 Write an equation to represent the problem, then solve. Be prepared to show your equation! The number of students in your class has decreased by 12% since September. How many students were there at the start if there are currently 19 students? 87

88 48 Choose the equation that represents the situation. A 15% discount. A x x = 0.85x B x + 1.5x = 2.5x C x 0.015x = 0.985x D x 0.15x = 0.85x 88

89 49 Write an equation to represent the problem, then solve. Be prepared to show your equation! When you paid your bill at a restaurant, you included 24% more to cover tax and tip. If you paid $55.80, what was the amount of the original bill? 89

90 Simple Interest Formula (Derived from ( Larry invests $100 in a savings plan. The plan pays 4.5% interest each year on his $100 account balance. The following chart shows the balance on his account after each year for the next 5 years. He did not make any deposits of withdrawals during this time. Time (in years) Balance (in dollars) What pattern(s) do you notice from the table? What is simple interest? How is it calculated? Can you create a formula to represent the pattern(s) you notice? 90

91 Simple Interest Formula (Derived from ( To find simple interest, use: Interest = Principal x Rate x Time I = P x r x t I = Prt r is the percent of the principal that is paid over a period of time (usually per year). t is the time. r and t must be compatible. For example, if r is an annual interest rate, then t must be written in years. 91

92 Simple Interest Formula (Derived from ( Can Money Grow? A Look at Simple Interest Larry invests $100 in a savings plan. The plan pays 4 1/2% interest each year on his $100 account balance. How much money will Larry earn in interest after 3 years? 5 years? 3 years: I = Prt I = 100(0.045)(3) I = years: I = Prt I = 100(0.045)(5) I = Larry will earn $22.50 in interest after 5 years. 92

93 Simple Interest Formula (Derived from ( How can you find the balance of Larry's account at the end of 5 years? : Add the interest earned after 5 years to the beginning balance. Click $ $100 = $

94 (Problem derived from 50 A $1,000 savings bond earns simple interest at the rate of 3% each year. The interest is paid at the end of every month. How much interest will the bond have earned after 3 months? ( 94

95 ( (Problem derived from 51 Mr. Williams wants to know how long it will take an investment of $450 to earn $200 in interest if the yearly interest rate is 6.5%, paid at the end of each year. 95

96 ( (Problem derived from 52 Find the amount of simple interest, A, earned on a $600 investment after 1 1/2 years if the semi annual (6 month) interest rate is 2%. 96

97 ( (Problem derived from 53 A $1,500 loan has an annual interest rate of 4 1/4% on the amount borrowed. How much time has elapsed if the interest is not $127.50? 97

98 Applied Percent of Decrease Return to table of contents 98

99 Applied Percent of Decrease There are situations when the percent of change is going to be a decrease. Examples are: Discounts Sales Reduction in Population 99

100 Applied Percent of Decrease When finding a discount, there are two different methods you can use. Method 1: Find the percent of the original price (discounted amount in $) Subtract the discount from the original price. Method 2: Subtract the percent from 100% (percent you are paying) Find the percent of the original price. 100

101 Applied Percent of Decrease Example: A $50 sweater is on sale for 20% off. Calculate the sale price. Method 1: Find the percent of the original price (discounted amount in $) Subtract the discount from the original price. (Discount) (Sale price) Method 2: Subtract the percent from 100% (percent you are paying) Find the percent of the original price. (Percent you pay) (Sale price) 101

102 Applied Percent of Decrease A manager wants to provide a 30% discount for everything in his store. Find the sale price of a $25 sweater. (Discount) (Percent you pay) (Sale price) (Sale price) Click to view Method 1 Click to view Method 2 Using either method, the Click answer to view is $17.50 answer 102

103 Applied Percent of Decrease The manager has pants, priced at $45, that he needs to mark down 35%. What will be the sale price of the pants? (Discount) (Percent you pay) (Sale price) (Sale price) Click to view Method 1 Click to view Method 2 The pants are on sale for Click $29.25 to view answer 103

104 Applied Percent of Decrease Mark wants to purchase a stereo that is on sale, if he is saving at least 30%. The stereo's original cost is $425. What is the most that he is willing to pay for the stereo? (Discount) (Percent you pay) (Sale price) (Sale price) Click to view Method 1 Click to view Method 2 He is willing to pay $ for the stereo. Click to view answer 104

105 54 Decrease 400 by 10% 105

106 55 A $710 computer is to be discounted 30%. What will be the sale price? 106

107 56 A necklace, priced at $120, is to be marked down 15%. What will be the sale price? 107

108 57 The student population of the high school will decrease by 5% next year. The current population is 1407 students. How many students will attend next year? 108

109 58 The store is having a 40% off sale. What percent will the customers pay? 109

110 59 $80 boots are on sale for 20% off. After the sale, the manager raises the price 20%. What will be the selling price of the boots after the sale? 110

111 Applied Percent of Increase Return to table of contents 111

112 Applied Percent of Increase There are situations when the percent of change is going to be an increase. Examples are: Tips Sales Tax Increase in Population 112

113 Applied Percent of Increase When finding an increase, there are two different methods you can use. Method 1: Find the percent of the original price (increased amount) Add the increase to the original price. Method 2: Add the percent to 100% (percent you are paying) Find the percent of the original price. 113

114 Applied Percent of Increase Increase 55 by 20% Finding a New Amount (Mark up) (New cost) (New cost) (Percent you pay) 114

115 Applied Percent of Increase Find the new amount Increase 60 by 10% Increase 68 by 12% 115

116 60 Increase 36 by 25%. 116

117 61 Increase 40 by 15% 117

118 Tip Tip: An amount added to a bill for services provided. Customers traditionally tip 18 20% for good service in restaurants and salons. Example: If the restaurant bill is $45 and you want to leave a 20% tip, how much money should you leave? (45) = 54 or 45(1.20) = 54 The customer will leave $54 on the table. The waitress will receive a $9 tip and the restaurant will receive $45. To calculate the amount of the tip only:.20(45) = 9 118

119 Calculate a 20% tip on a $75 bill. Tip What will the customer leave in total? For poor service, my friend will leave a 5% tip. How much less will this waitress earn than the waitress above? 119

120 Sales Tax Sales tax : An amount of money that is calculated by applying a percentage rate to the taxable price of a sale. Sales taxes are collected by the buyer from the seller, who turns it over to the government. In NJ the sales tax rate is 7%. To calculate Sales Tax alone find the percent (tax) of the price. That is the amount that you owe in addition to the cost of the item. To find the total cost of an item, you must add the sales tax to the cost. There are 2 ways to do this: 1. Find the percent of the item and add it to the original amount. 2. Find 100% + tax% of the original amount. 120

121 Sales Tax A car costs $23,500. How much sales tax will the customer pay? 23,500(0.07) = $1645 What will the customer pay altogether for the car? 23, = $25,145 The total cost of the car, including tax, can be calculated as follows: 23, (23,500) = 25,145 or 23,500(1.07) = 25,

122 Discuss How are tips and sales tax alike? 122

123 62 What is the total cost of a $250 stereo in the state of NJ? 123

124 63 Calculate the sales tax on a $125 bicycle. 124

125 64 Mike wants to leave a 20% tip. His bill is $ How much is the tip? 125

126 65 A $65 restaurant tab is put on the table. The couple plans on leaving an 18% tip. How much should be left altogether? 126

127 Real Life Application Problems Return to table of contents 127

128 Application Problems A store owner pays $12 for a particular bracelet. To cover expenses, the owner will mark up the price by 150%. Find the selling price of the bracelet. 128

129 Application Problems The store is having a 20% off sale on all CD's. With the sale, you pay $12 for a CD. What was the original price? A couple left their waiter a 20% tip in the amount of $18. What was the cost of their meal? 129

130 Application Problems The store is having a 20% off sale on all CD's. With the sale, you pay $12 for a CD. What was the original price? 130

131 Application Problems A couple left their waiter a 20% tip in the amount of $18. What was the cost of their meal? 131

132 Application Problems You and 3 friends had dinner at a restaurant. The cost of their meals is $62. They want to leave a 15% tip. Calculate the tip. When they arrive at the register the cashier will calculate the sales tax on the meal at a rate of 7%. Determine the sales tax. (*Note: You never tax on the tip) Calculate the total cost of the meal for each of you. 132

133 Application Problems A store is having a 25% off sale on ipods. You want to purchase an ipod with an original price of $249. The sales tax is 7%, which will be applied to the sale price of the ipod. What is the total cost of the ipod? 133

134 Application Problems A computer is on sale for 10% off the original price of $325. When it doesn't sell, the manager marks it down another 20% off the sale price. What is the new sale price of the laptop? Is the new sale price the same as it would be had the manager taken 30% off of the original price? Explain. 134

135 66 Wholesale price: $56 Markup percent: 50% New price? 135

136 67 Tickets cost $7 at the door. If purchased in advance, the tickets cost $5. What is the percent of discount for purchasing tickets in advance? 136

137 68 Five hundred sixty people were surveyed. 25% said they prefer Coke. How many people prefer Coke? 137

138 69 Increase 50 by 25%. What is the new amount? 138

139 70 What is the original price on a pair of boots that sell for $72 after a 25% discount? 139

140 71 An ipod costs $176. It is on sale for 20% off and will be taxed at a rate of 7% on the sale price. What will be the total cost of the ipod? 140

141 72 What is the total cost of a $123 ipod, including tax? 141

142 A teacher survey students in four classes to determine the location for a field trip. Each student chose only one location. The table shows the number of students from each class who chose each location. (Use this table for the next two questions.) From PARCC PBA sample test calculator #4 142

143 Part A Determine the percent of students in each class who chose the museum. What is the order from greatest to least of the percents for each class? Drag and drop the classes into the correct order from greatest to least with the greatest at the top. Class E Class F Class G Class H From PARCC sample test 143

144 73 Part B The total number of students who chose the zoo is how many times as great as the total number of students who chose the planetarium? From PARCC sample test 144

145 74 A store owner paid $15 for a book. She marked up the price of the book by 40% to determine its selling price. Part A What is the selling price of the book? From PARCC EOY sample test calculator #10 145

146 75 Part B A customer buys a different book that has an original selling price of $38. The book is discounted 25%. The customer must pay a 6% sales tax on the discounted price of the book. What is the total amount the customer pays for the discounted book? From PARCC EOY sample test calculator #10 146

147 76 The students in Noami's class sold calendars for a fundraiser this year and last year. This year, the selling price of each calendar was $ The price this year represents 6% more than the selling price of each calendar last year. Part A What is the selling price of each calendar last year? From PARCC EOY sample test calculator #6 continued

148 77 Part B The students in Naomi's class earned 20% of the selling price of each calendar sold this year and last year. At last year's selling price, Naomi's class sold 650 calendars. At this year's selling price, Naomi's class sold 600 calendars. Select a choice from each group to fill in the blanks. The students in Naomi's class earned more money from the fundraiser by. C $20 A last year D $25 E $35 B this year F $50 G $60 From PARCC EOY sample test calculator #6 148

149 78 Each bulleted statement describes how the amount of income tax is determined for yearly incomes in different ranges. Yearly incomes of 8,925 or less are taxed at a flat rate of 10%. For yearly incomes from $8,926 to $36,250, the first $8,925 is taxed at 10% and any income beyond that is taxed at 15%. For yearly incomes greater than $36.250, the first $8,925 is taxed at 10%, the next $27,325 is taxed at 15% and any income beyond $36,250 is taxed at 25%. Part A Mr. Vance's yearly taxable income is $35,675. What is the dollar amount taken our for taxes based on Mr. Vance's income? continued... From PARCC EOY sample test calculator #9 149

150 79 Part B Mr. Rivera's taxable income is $20 each hour before taxes are taken out. Mr. Rivera worked a total of 40 hours each week for 50 weeks. What is the dollar amount taken out for taxes based on Mr. Rivera's taxable income? From PARCC EOY sample test calculator #9 150

151 Glossary Teacher Notes Return to Table of Contents 151

152 "Is" Equals The "part" in a proportion when computing percentages. is What is 15% of 90? Back to Instruction 152

153 "Of" Multiply The "whole" in a proportion when computing percentages is 15% of what number? of Back to Instruction 153

154 Proportion A statement that two ratios (fractions) are equal to each other. When computing a percentage: or Back to Instruction 154

155 Percent of Change The ratio of the amount of increase or decrease to the original amount. Amount of increase or decrease = % Original Amount 100 Increas e new amount > original new amount Decreas e amount < original amount original amount: 20 new amount: 30 Back to Instruction 155

156 Tip An amount added to a bill for services provided. Customers traditionally tip 18 20% for good service in restaurants and salons. 20% tip on $45 bill:.20(45) = $9 tip How much money will you leave on the table? $45 bill + $9 = $54 total Back to Instruction 156

157 Sales Tax An amount of money that is calculated by applying a percentage rate to the taxable price of a sale. In NJ the sales tax rate is 7%. A car costs $23,500. What is the amount of the sales tax? $23,500(0.07) = $1,645 How much money will you pay total? $23,500 + $1,645 = $25,145 Back to Instruction 157