7th Grade. Relating Fractions, Decimals & Percents. Slide 1 / 157 Slide 2 / 157. Slide 3 / 157. Slide 4 / 157. Slide 6 / 157. Slide 5 / 157.

Transcription

1 Slide 1 / 157 Slide 2 / 157 7th Grade Percents Slide 3 / 157 Table of Contents Slide 4 / 157 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 Applied Percent of Increase Real-life Application Problems Glossary Relating Fractions, Decimals & Percents Return to table of contents Slide 5 / 157 Helping you remember... Fill in each box below with an example of the process described. Slide 6 / 157 Ordering Order the numbers from least to greatest % % to a fraction % to a decimal fraction to a % decimal to a % 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% % 0.15

2 1 Find the lowest value. Slide 7 / Find the greatest value. Slide 8 / 157 A 5% B 1/2 C.5% D.05 A 120% B 1.02 C.2% D Find the greatest value. Slide 9 / Find the lowest value. Slide 10 / 157 A 6% B.6 C 60 D 6 A 2% B.2 C.02 D.2% Slide 11 / 157 Slide 12 / Find the lowest value. A 50% B 500% C 50.0 D 50.01

3 Slide 13 / 157 Examples 6 Express as a fraction. Slide 14 / 157 Express each fraction as a percent: 1) 2) 3) click click click 7 Express as a decimal. Slide 15 / Express as a percent. Slide 16 / Express as a decimal. Slide 17 / Express as a percent. Slide 18 / 157

4 Slide 19 / 157 Slide 20 / Express as a percent. Three Types of Percent Problems Return to table of contents Slide 21 / 157 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? Slide 22 / 157 Two words that will occur in these types of problems are: "is" "of" These words have specific meanings in math. "Is" means equals (=) "Of" means multiply Types of Percent Problems 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. Slide 23 / 157 Slide 24 / 157 Find the Part Finding the Part... Examples: Find 40% of = 24 Click Write a mathematical sentence 20% of = 18 Click Write a mathematical sentence

5 Slide 25 / 157 Slide 26 / 157 Write a Mathematical Sentence What is 10% of 88? X = X = 8.8 Try these: Find 12% of 70 What is 40% of 28? Slide 27 / 157 Slide 28 / 157 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. Slide 29 / 157 Slide 30 / 157 Proportion Method Example Proportion Method Example Example: What is 25% of 400? Example: What is 32% of 300? Steps 1. Set up the proportion. is of? 400 = % 100 Steps 1. Set up the proportion. is of? 300 = % Substitute. 3. Solve. What is 25% of 400? click 400 x 25 = 100w 10,000 = 100w 10,000/100 = w 100 = w Click on each box to see if you substituted correctly. 2. Substitute. What is 32% of 300? 3. Solve. click 300 x 32 = 100w 9,600 = 100w 9600/100 = w 96 = w Click on each box to see if you substituted correctly.

6 Slide 31 / 157 Slide 32 / 157 Proportion Method - Try It 12 Find 30% of 45. Try it: What is 20% of 180? Steps 1. Set up the proportion. is of = % Substitute. 3. Solve. Slide 33 / 157 Slide 34 / What is 15% of 90? 14 Find the greater value. A 20% of 16 B 10% of 90 C 25% of 40 D 100% of 7 15 Find the greater value. A 2% of 1000 B 5% of 500 C 10% of 300 D 15% of 100 Slide 35 / 157 Slide 36 / Identify any values that are equal. A What is 40% of 80? B 60% of 70 C 25% of 128 D 200% of 16

7 Slide 37 / 157 Finding the Whole... 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 Slide 38 / 157 Finding the Whole X = X = 125 Try This: 100 is 20% of what number? 100 =.20 x Slide 39 / 157 Finding the Whole is 70% of what? Slide 40 / = x.20 x = % of what number is 6? Slide 41 / 157 Slide 42 / % of what number is 10?

8 Slide 43 / is 150% of what number? 21 1% of what number is 12? Slide 44 / 157 Slide 45 / 157 Slide 46 / 157 Finding the Percent 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% 60 is what percent of 15? 60 = x 15 Slide 47 / 157 Finding the Percent 22 What percent of 3 is 12? Slide 48 / = X 15 4 = X 400% = X

9 23 30 is what percent of 36? Slide 49 / 157 Slide 50 / What percent of 18 is 180? 25 2 is what percent of 1? Slide 51 / What percent of 25 is 20? Slide 52 / 157 Slide 53 / 157 You have just studied three different types of percent problems. Try all 3 types: 24 is 40% of what number? Percent Problems Slide 54 / 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? 42 is what percent of 840? What is 30% of 45?

10 Slide 55 / 157 Slide 56 / Find the greatest percentage value. 29 Find 20% of 78. 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? Slide 57 / Eight is what percent of 28? Slide 58 / What number is 3% of 17? 32 Find 27% of 54. Slide 59 / 157 Slide 60 / is what percent of 200?

11 34 What percent is 35 of 20? Slide 61 / 157 Slide 62 / Fifty six percent of what number is 40? Slide 63 / Forty five is 30% of what number? Slide 64 / Sixty two percent of 40 is what number? A 24.8 B.0155 C 24.8% D 15.5 Slide 65 / 157 Slide 66 / 157 Percent of Change Percent of Change is the ratio of the amount of increase or decrease to the original amount. Percent of Change 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 100 Return to table of contents

12 Slide 67 / 157 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 Slide 68 / 157 Identify the percent of change as an increase or decrease. Then find the percent of change. 1. Original: 45 New: 75 Percent of Change Percent of change= Percent of change= 2. Original: 100 New: Original: 58 New: 75 Slide 69 / 157 Percent of Change Slide 70 / 157 Percent of Change Try This! A CD's original price was $ It is now on sale for $ What is the 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? Slide 71 / 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? Slide 72 / Original Amount: 500 New: 700 Find the percent of change.

13 Slide 73 / Original Amount: 52 New: 17 Find the percent of change. Slide 74 / 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? Slide 75 / Find the percent of change. Original price: $120 Sale price: $75 Slide 76 / Find the percent of change. Original price: $80 Sale price: $50 Slide 77 / 157 Slide 78 / A stereo, originally priced at $360, is on sale for $200. What is the percent of change? Representing Percent Equations Algebraically Return to table of contents

14 Slide 79 / 157 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. 100% + 5% = 105% Slide 80 / 157 Think about this... 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 Example: Slide 81 / 157 Representing Percent Equations Algebraically 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 $ Example: Slide 82 / 157 Representing Percent Equations Algebraically 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 Slide 83 / 157 Representing Percent Equations Algebraically So, what does this mean? m m = 1.15m This could mean increase m by 15% or multiply m by Slide 84 / 157 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. They mean the same thing! Likewise, what is the meaning of w w = 0.58w Click 2. What does this equation mean? p p = 1.02p 3. What does this equation mean? h - 0.1h = 0.9h This means both decrease w by 42% or multiply w by Click

15 Slide 85 / 157 Slide 86 / 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? 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 x = 0.99x Slide 87 / 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? Slide 88 / Choose the equation that represents the situation. A 15% discount. A x x = 0.85x B x + 1.5x = 2.5x C x x = 0.985x D x x = 0.85x Slide 89 / 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? Slide 90 / 157 Simple Interest Formula 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) (Derived from ( 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?

16 Slide 91 / 157 Slide 92 / 157 Simple Interest Formula (Derived from ( 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. Slide 93 / 157 Simple Interest Formula (Derived from How can you find the balance of Larry's account at the end of 5 years? ( 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: 5 years: I = Prt I = 100(0.045)(3) I = I = Prt I = 100(0.045)(5) I = Larry will earn $22.50 in interest after 5 years. Slide 94 / 157 (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? ( Answer: Add the interest earned after 5 years to the beginning balance. Click $ $100 = $ Slide 95 / 157 Slide 96 / 157 (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. ( (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%. (

17 Slide 97 / 157 Slide 98 / 157 (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? ( Applied Percent of Decrease Return to table of contents Slide 99 / 157 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 Slide 100 / 157 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. Slide 101 / 157 Slide 102 / 157

18 Slide 103 / 157 Slide 104 / Decrease 400 by 10% Slide 105 / 157 Slide 106 / A $710 computer is to be discounted 30%. What will be the sale price? Slide 107 / A necklace, priced at $120, is to be marked down 15%. What will be the sale price? Slide 108 / 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?

19 Slide 109 / The store is having a 40% off sale. What percent will the customers pay? Slide 110 / $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? Slide 111 / 157 Slide 112 / 157 Applied Percent of Increase There are situations when the percent of change is going to be an increase. Examples are: Applied Percent of Increase Tips Sales Tax Increase in Population Return to table of contents Slide 113 / 157 Slide 114 / 157 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.

20 Slide 115 / 157 Applied Percent of Increase 60 Increase 36 by 25%. Slide 116 / 157 Find the new amount Increase 60 by 10% Increase 68 by 12% 61 Increase 40 by 15% Slide 117 / 157 Slide 118 / 157 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 Calculate a 20% tip on a $75 bill. Slide 119 / 157 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? Slide 120 / 157 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.

21 Slide 121 / 157 Sales Tax A car costs $23,500. How much sales tax will the customer pay? Slide 122 / 157 Discuss How are tips and sales tax alike? 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) = 25,145 Slide 123 / What is the total cost of a $250 stereo in the state of NJ? Slide 124 / Calculate the sales tax on a $125 bicycle. Slide 125 / Mike wants to leave a 20% tip. His bill is $ How much is the tip? Slide 126 / A $65 restaurant tab is put on the table. The couple plans on leaving an 18% tip. How much should be left altogether?

22 Slide 127 / 157 Slide 128 / 157 Application Problems Real Life 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. Return to table of contents Slide 129 / 157 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? Slide 130 / 157 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? Slide 131 / 157 Application Problems Slide 132 / 157 Application Problems A couple left their waiter a 20% tip in the amount of $18. What was the cost of their meal? 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.

23 Slide 133 / 157 Application Problems Slide 134 / 157 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? 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. 66 Wholesale price: $56 Markup percent: 50% New price? Slide 135 / 157 Slide 136 / 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? Slide 137 / Five hundred sixty people were surveyed. 25% said they prefer Coke. How many people prefer Coke? Slide 138 / Increase 50 by 25%. What is the new amount?

24 Slide 139 / What is the original price on a pair of boots that sell for $72 after a 25% discount? Slide 140 / 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? Slide 141 / What is the total cost of a $123 ipod, including tax? Slide 142 / 157 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 Part A Slide 143 / 157 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. 73 Part B Slide 144 / 157 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? Class E Class F Class G Class H From PARCC sample test From PARCC sample test

25 Slide 145 / 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? 75 Part B Slide 146 / 157 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 From PARCC EOY sample test calculator #10 Slide 147 / 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 Slide 149 / 157 continued 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 77 Part B Slide 148 / 157 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 79 Part B Slide 150 / 157 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

26 Slide 151 / 157 Slide 152 / 157 Glossary Return to Table of Contents Slide 153 / 157 Slide 154 / 157 Slide 155 / 157 Percent of Change The ratio of the amount of increase or decrease to the original amount. Amount of increase or decrease = % Original Amount 100 Slide 156 / 157 Tip An amount added to a bill for services provided. Increas e new amount original new > amount Decreas e amount < original amount original amount: 20 new amount: 30 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 Back to Instruction

27 Slide 157 / 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