Ratios, Rates & Proportions

Transcription

1 Slide 1 / 130 Ratios, Rates & Proportions Table of Contents Click on the topic to go to that section Slide 2 / 130 Writing Ratios Equivalent Ratios Rates Writing an Equivalent Rate Proportions Application problems Sampling Scale Drawings Similar Figures Slide 3 / 130 Writing Ratios Return to Table of Contents

2 Ratios Slide 4 / 130 What do you know about ratios? When have you seen or used ratios? Ratio - A comparison of two numbers by division Find the ratio of boys to girls in this class Slide 5 / 130 Ratios can be written three different ways: a to b a : b a b Each is read, "the ratio of a to b." Each ratio should be in simplest form. There are 48 animals in the field. Twenty are cows and the rest are horses. Slide 6 / 130 Write the ratio in three ways: a. The number of cows to the number of horses b. The number of horses to the number of animals in the field Remember to write your ratios in simplest form!

3 1 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of vanilla cupcakes to strawberry cupcakes? Slide 7 / 130 A 7 : 9 B 7 27 C 7 11 D 1 : 3 2 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate & strawberry cupcakes to vanilla & chocolate cupcakes? Slide 8 / 130 A B 11 7 C 5 4 D There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate cupcakes to total cupcakes? Slide 9 / 130 A 7 9 B 7 27 C 9 27 D 1 3

4 4 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of total cupcakes to vanilla cupcakes? Slide 10 / 130 A 27 to 9 B 7 to 27 C 27 to 7 D 11 to 27 Slide 11 / 130 Equivalent Ratios Return to Table of Contents Equivalent ratios have the same value Slide 12 / : 2 is equivalent to 6: 4 1 to 3 is equivalent to 9 to is equivalent to 42

5 There are two ways to determine if ratios are equivalent x x 3 Since the numerator and denominator were multiplied by the same value, the ratios are equivalent Slide 13 / Cross Products Since the cross products are equal, the ratios are equivalent. 4 x 15 = 5 x = is equivalent to Slide 14 / 130 True False 6 5 is equivalent to Slide 15 / 130 True False

6 7 18:12 is equivalent to 9, which is equivalent to Slide 16 / 130 True False 8 2 is equivalent to 10, which is equivalent to Slide 17 / 130 True False 9 1:7 is equivalent to 10, which is equivalent to 5 to Slide 18 / 130 True False

7 Slide 19 / 130 Rates Return to Table of Contents Rate: a ratio of two quantities measured in different units Slide 20 / 130 Examples of rates: 4 participants/2 teams 5 gallons/3 rooms 8 burgers/2 tomatoes Unit rate: Rate with a denominator of one Often expressed with the word "per" Examples of unit rates: 34 miles/gallon 2 cookies per person 62 words/minute Finding a Unit Rate Slide 21 / 130 Six friends have pizza together. The bill is $63. What is the cost per person? Hint: Since the question asks for cost per person, the cost should be first, or in the numerator. $63 6 people Since unit rates always have a denominator of one, rewrite the rate so that the denominator is one. $ people 6 $ person The cost of pizza is $10.50 per person

8 10 Sixty cupcakes are at a party for twenty children. How many cupcakes per person? Slide 22 / John's car can travel 94.5 miles on 3 gallons of gas. How many miles per gallon can the car travel? Slide 23 / The snake can slither 24 feet in half a day. How many feet can the snake move in an hour? Slide 24 / 130

9 13 There are six chaperones at the dance of 100 students. How many students per chaperone are there? Slide 25 / The recipe calls for 6 cups of flour for every four eggs. How many cups of flour are needed for one egg? Slide 26 / 130 Population Density Slide 27 / 130 Population Density: A unit rate of people per square mile This data is compiled by the US Census Bureau every 10 years and is used when determining the number of Representatives each state gets in the House of Representatives. Look at the following Population Densities from New Jersey: 1,184 people per square mile Montana: 7 people per square mile California: 237 people per square mile

10 15 The population of Newark, NJ is 278,980 people in square miles. What is its population density? Slide 28 / The population of Moorestown, NJ is 19,509 people in 15 square miles. What is its population density? Slide 29 / The population of Waco, TX is 124,009 people in 75.8 square miles. What is its population density? Slide 30 / 130

11 Slide 31 / 130 Writing an Equivalent Rate Return to Table of Contents Slide 32 / 130 To write equivalent rates, conversion factors must be used. Conversion factors are used to convert from one unit to another. Conversion factors must be equal to 1. Some examples of conversion factors: 1 pound or 16 ounces 16 ounces 1 pound 12 inches or 1 foot 1 foot 12 inches 3 feet or 1 yard 1 yard 3 feet 1 day or 24 hours 24 hours 1 day Create 5 conversion factors of your own! Identify the conversion factor that results in the desired unit. Slide 33 / 130 Find a conversion factor that converts minutes to seconds. minutes seconds PULL 60 seconds 1 minute or 1 minute 60 seconds Hint: You want the rate of minute to cancel, so that you are left with the rate of seconds

12 Identify the conversion factor that results in the desired unit. Slide 34 / 130 Find a conversion factor that converts 12 feet to yards. 12 feet? yards PULL 3 feet 1 yard or 1 yard 3 feet PULL Hint: You want the rate of feet to cancel, so that you are left with the rate of yards. Identify the conversion factor that results in the desired unit. Slide 35 / 130 Find a conversion factor that converts miles to feet. 5 miles? feet PULL 5280 feet 1 mile or 1 mile 5280 feet Hint: You want the rate of miles to cancel, so that you are left with the rate of feet PULL Slide 36 / 130 To write equivalent rates, conversion factors must be used. Example 1: 2 inches? inches 1 hour 1 day 2 inches 24 hours 48 inches 1 hour 1 day 1 day Example 2: 5 feet? feet 1 sec 1 hour 5 feet 60 sec 300 feet 1 sec 1 hour 1 hour

13 18 Write the equivalent rate. 40 mi? mi 1 min 1 h Slide 37 / Write the equivalent rate. 54 inches? inches 1 year 1 month Slide 38 / Write the equivalent rate. 1 day 1week $75? dollars Slide 39 / 130

14 21 Write the equivalent rate. 30 sec 1min 425 mi? miles Slide 40 / Write the equivalent rate. 40 feet inches 3 hrs hr Slide 41 / 130 Hint: Find the equivalent rate and then determine the unit rate 23 Write the equivalent rate. 20,000 feet? feet 4 seconds minute Slide 42 / 130 Hint: Find the equivalent rate and then determine the unit rate

15 24 Write the equivalent rate people? people 6 days hr Slide 43 / 130 Hint: Find the equivalent rate and then determine the unit rate Slide 44 / 130 Proportions Return to Table of Contents A proportion is an equation that states that two ratios are equivalent. Slide 45 / 130 Example:

16 If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Slide 46 / 130 Example 1: x x x Hint: To find the value of x, multiply 3 by 3 also x 3 If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Slide 47 / 130 Example: x x Hint: To find the value of x, divide 32 by 4 also Solve the proportion using equivalent ratios Slide 48 / x

17 26 Solve the proportion using equivalent ratios 4 x 9 36 Slide 49 / Solve the proportion using equivalent ratios Slide 50 / x 28 Solve the proportion using equivalent ratios Slide 51 / 130 x

18 29 Solve the proportion using equivalent ratios Slide 52 / x 28 In a proportion, the cross products are equal. Slide 53 / Proportions can also be solved using cross products. Slide 54 / x Cross multiply 4x = x = 60 x = 15 Solve for x Example 2 7 x 8 48 Cross multiply 8x = x = 336 x = 42 Solve for x

19 30 Use cross products to solve the proportion 9 = x Slide 55 / Use cross products to solve the proportion x = Slide 56 / Use cross products to solve the proportion 45 = _x 18 6 Slide 57 / 130

20 33 Use cross products to solve the proportion 2 = _x Slide 58 / Use cross products to solve the proportion 7 = _3 x 21 Slide 59 / 130 Slide 60 / 130 Application problems Return to Table of Contents

21 Chocolates at the candy store cost $5.99 per dozen. How much does one candy cost? Round your answer to the nearest cent. Slide 61 / 130 Solution: $ dozen 1 dozen 12 (Use equivalent rates) $ $0.50 per candy Example 2: Slide 62 / 130 There are 3 books per student. There are 570 students. How many books are there? Set up the proportion: Books Students 3 Where does the 570 go? 1 3 x x x 1,710 books Example 3: Slide 63 / 130 The ratio of boys to girls is 4 to 5. There are 125 people on a team. How many are girls? Set up the proportion: Girls People How did we determine this ratio? 5 Where does the 125 go? = 9 5 = x x = x = 625 x = girls

22 35 Cereal costs $3.99 for a one pound box. What is the price per ounce? Round your answer to the nearest penny. Slide 64 / Which is the better buy? Brand A: $2.19 for 12 ounces Brand B: $2.49 for 16 ounces Slide 65 / 130 A B Brand A Brand B 37 There are 4 girls for every 10 boys at the party. There are 56 girls at the party. How many boys are there? Slide 66 / 130

23 38 The farmer has cows and chickens. He owns 5 chickens for every cow. He has a total of 96 animals. How many cows does he own? Slide 67 / The auditorium can hold 1 person for every 5 square feet. It is 1210 square feet. How many people can the auditorium hold? Slide 68 / The recipe for one serving calls for 4 oz of beef and 2 oz of bread crumbs. 50 people will be attending the dinner. How many lbs. of bread crumbs should be purchased? Slide 69 / 130

24 41 Mary received 4 votes for every vote that Jane received people voted. How many votes did Jane receive? Slide 70 / To make the desired shade of pink paint, Brandy uses 3 oz. of red paint for each oz. of white paint. She needs one quart of pink paint. How many oz. of red paint will she need? Slide 71 / In a sample of 50 randomly selected students at a school, 38 students eat breakfast every morning. There are 652 students in the school. Using these results, predict the number of students that eat breakfast. Slide 72 / 130 A 76 B 123 C 247 D 496 Question from ADP Algebra I End-of-Course Practice Test

25 Slide 73 / 130 Sampling Return to Table of Contents Slide 74 / 130 Your task is to count the number of whales in the ocean or the number of squirrels in a park. How could you do this? What problems might you face? Slide 75 / 130 How would you estimate the size of a crowd? What methods would you use? Could you use the same methods to estimate the number of wolves on a mountain?

26 One way to estimate the number of wolves on a mountain is to use the CAPTURE - RECAPTURE METHOD. Slide 76 / 130 Suppose this represents all the wolves on the mountain. Slide 77 / 130 Wildlife biologists first find some wolves and tag them. Slide 78 / 130

27 Then they release them back onto the mountain. Slide 79 / 130 They wait until all the wolves have mixed together. Then they find a second group of wolves and count how many are tagged. Slide 80 / 130 Slide 81 / 130 Biologists use a proportion to estimate the total number of wolves on the mountain: tagged wolves on mountain total wolves on mountain = tagged wolves in second group total wolves in second group For accuracy, they will often conduct more than one recapture. 8 2 = w 9 2w = 72 w = 36 There are 36 wolves on the mountain

28 Try This: Biologists are trying to determine how many fish are in the Rancocas Creek. They capture 27 fish, tag them and release them back into the Creek. 3 weeks later, they catch 45 fish. 7 of them are tagged. How many fish are in the creek? Slide 82 / = f 45 27(45) = 7f 1215 = 7f = f There are 174 fish in the river A whole group is called a POPULATION. Slide 83 / 130 A part of a group is called a SAMPLE. When biologists study a group of wolves, they are choosing a sample. The population is all the wolves on the mountain. Population Sample Example: Slide 84 / out of 4,000 people surveyed watched Grey's Anatomy. How many people in the US watched if there are 93.1 million people? 860 = x ,100, (93,100,000) = 4000x 80,066,000,000 = 4000x 20,016,500 = x 20,016,500 people watched

29 Try This: Slide 85 / out of 600 people surveyed voted for Candidate A. How many votes can Candidate A expect in a town with a population of 1500? Margin of Error Slide 86 / 130 The results of sampling are estimates, which always contain some error. The margin of error estimates the interval that is most likely to include the exact result for the population. Margin of error is given as a percent in the problem. To find the interval using margin of error: Find the percent of the population Add/Subtract that amount from the answer to create an interval. Slide 87 / 130

30 Slide 88 / You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Slide 89 / You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Use a 2% margin of error. Slide 90 / 130 What is the amount you are going to + by?

31 46 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Use a 2% margin of error. Slide 91 / 130 What is the lower number in your interval? 47 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Use a 2% margin of error. Slide 92 / 130 What is the upper number in your interval? 48 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Slide 93 / 130

32 49 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Find an interval using a 3% margin of error. Slide 94 / 130 What is the amount you are going to + by? 50 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Find an interval using a 3% margin of error. Slide 95 / 130 What is the lower number in your interval? 51 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Find an interval using a 3% margin of error. Slide 96 / 130 What is the upper number in your interval?

33 Slide 97 / 130 Scale Drawings Return to Table of Contents Scale drawings are used to represent objects that are either too large or too small for a life size drawing to be useful. Slide 98 / 130 Examples: A life size drawing of an ant or an atom would be too small to be useful. A life size drawing of the state of New Jersey or the Solar System would be too large to be useful. A scale is always provided with a scale drawing. Slide 99 / 130 The scale is the ratio: drawing real life (actual) When solving a problem involving scale drawings you should: Write the scale as a ratio Write the second ratio by putting the provided information in the correct location (drawing on top & real life on the bottom) Solve the proportion

34 Example: Slide 100 / 130 This drawing has a scale of "1:10", so anything drawn with the size of "1" would have a size of "10" in the real world, so a measurement of 150mm on the drawing would be 1500mm on the real horse. Example: Slide 101 / 130 The distance between Philadelphia and San Francisco is 2,950 miles. You look on a map and see the scale is 1 inch : 100 miles. What is the distance between the two cities on the map? Write the scale as a ratio 1 x 100 = x = 2950 x = inches on the map Try This: Slide 102 / 130 On a map, the distance between your town and Washington DC is 3.6 inches. The scale is 1 inch : 55 miles. What is the distance between the two cities?

35 52 The distance between Moorestown, NJ and Duck, NC is 910 miles. What is the distance on a map with a scale of 1 inch to 110 miles? Slide 103 / The distance between Philadelphia and Las Vegas is 8.5 inches on a map with a scale 1.5 in : 500 miles. What is the distance in miles? Slide 104 / You are building a room that is 4.6 m long and 3.3 m wide. The scale on the architect's drawing is 1 cm : 2.5 m. What is the length of the room on the drawing? Slide 105 / 130

36 55 You are building a room that is 4.6 m long and 3.3 m wide. The scale on the architect's drawing is 1 cm : 2.5 m. What is the width of the room on the drawing? Slide 106 / Find the length of a 72 inch wide door on a scale drawing with a scale 1 inch : 2 feet. Slide 107 / You recently purchased a scale model of a car. The scale is 1cm : 24m. What is the length of the model car if the real car is 4m? Slide 108 / 130

37 58 You recently purchased a scale model of a car. The scale is 1cm : 24m. The length of the model's steering wheel is 2.25 cm. What is the actual length of the steering wheel? Slide 109 / The figure is a scale of the east side of a house. In the drawing, the side of each square represents 4 feet. Find the width and height of the door. Slide 110 / 130 A B C D 4 ft by 9 ft 4 ft by 12 ft 4 ft by 8 ft 4 ft by 10 ft 60 On a map, the scale is 1/2 inch= 300 miles. Find the actual distance between two stores that are 5 1/2 inches apart on the map. Slide 111 / 130 A B C D 3000 miles 2,727 miles 3,300 miles 1,650 miles

38 61 On a map with a scale of 1 inch =100 miles, the distance between two cities is 7.5 inches. If a car travels 55 miles per hour, about how long will it take to get from one city to the other. Slide 112 / 130 A B C D 13 hrs 45 min. 14 hrs 30 min. 12 hrs 12 hrs 45 min. Slide 113 / 130 Similar Figures Return to Table of Contents Two objects are similar if they are the same shape but different sizes. Slide 114 / 130 In similar objects: corresponding angles are congruent corresponding sides are proportional

39 To check for similarity: Slide 115 / 130 Check to see that corresponding angles are congruent Check to see that corresponding sides are proportional (Cross products are equal) Example: Slide 116 / 130 Is the pair of polygons similar? Explain your answer. 4 yd 3 yd 6 yd 4.5 yd 4 3 = (4.5) = 6(3) 18 = 18 YES Example: Slide 117 / 130 Is the pair of polygons similar? Explain your answer. 8 m 5 m 10 m 13 m 5 8 = (13) = 10(8) 65 = 80 NO

40 Example: Slide 118 / 130 Find the value of x in the pair of similar polygons. 15 cm x 6 cm 10 cm 8 cm 15 6 = x 10 15(10) = 6x 150 = 6x 25 cm = x Try This: Slide 119 / 130 Find the value of y in the pair of similar polygons. 15 in 7.5 in y 5 in 62 Are the polygons similar? You must be able to justify your answer. Slide 120 / 130 Yes No 15 ft 9 ft 21 ft 12 ft

41 63 Are the polygons similar? You must be able to justify your answer. Slide 121 / 130 Yes No 10 m 8 m 2.5 m 2 m 64 Are the polygons similar? You must be able to justify your answer. Slide 122 / 130 Yes No 15 yd 37.5 yd 15 yd 6 yd 65 Find the measure of the missing value in the pair of similar polygons. Slide 123 / y

42 66 Find the measure of the missing value in the pair of similar polygons. Slide 124 / ft 25 ft 25 ft w 18 ft 67 Find the measure of the missing value in the pair of similar polygons. Slide 125 / 130 x 17 m 4 m 4.25 m 68 Find the measure of the missing value in the pair of similar polygons. 6 mm y Slide 126 / mm 38.5 mm

43 69 Find the measure of the missing value in the pair of similar polygons. Slide 127 / m 13 m 7 m? 119 m 70 Find the measure of the missing value in the pair of similar polygons. Slide 128 / m? 7 m 13 m 9 m 119 m 71 Find the measure of the missing value in the pair of similar polygons. Slide 129 / mm 5 mm 27.5 mm x

44 Slide 130 / 130