Slide 1 / 130 Ratios, Rates & Proportions
Slide 2 / 130 Table of Contents Click on the topic to go to that section 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
Slide 4 / 130 Ratios 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.
Slide 6 / 130 There are 48 animals in the field. Twenty are cows and the rest are horses. 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!
Slide 7 / 130 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? A 7 : 9 B 7 27 C 7 11 D 1 : 3
Slide 8 / 130 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? A 20 16 B 11 7 C 5 4 D 16 20
Slide 9 / 130 3 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? A 7 9 B 7 27 C 9 27 D 1 3
Slide 10 / 130 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? 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
Slide 12 / 130 Equivalent ratios have the same value 3 : 2 is equivalent to 6: 4 1 to 3 is equivalent to 9 to 27 5 35 6 is equivalent to 42
4 12 5 15 x 3 Slide 13 / 130 There are two ways to determine if ratios are equivalent. 1. 4 12 5 15 x 3 2. Cross Products Since the numerator and denominator were multiplied by the same value, the ratios are equivalent 4 12 5 15 Since the cross products are equal, the ratios are equivalent. 4 x 15 = 5 x 12 60 = 60
Slide 14 / 130 5 4 is equivalent to 8 9 18 True False
Slide 15 / 130 6 5 is equivalent to 30 9 54 True False
Slide 16 / 130 7 18:12 is equivalent to 9, which is equivalent to 36 6 24 True False
Slide 17 / 130 8 2 is equivalent to 10, which is equivalent to 40 24 120 480 True False
Slide 18 / 130 9 1:7 is equivalent to 10, which is equivalent to 5 to 65 70 True False
Slide 19 / 130 Rates Return to Table of Contents
Slide 20 / 130 Rate: a ratio of two quantities measured in different units 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
Slide 21 / 130 Finding a Unit Rate 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. $63 6 6 people 6 $10.50 1 person The cost of pizza is $10.50 per person
Slide 22 / 130 10 Sixty cupcakes are at a party for twenty children. How many cupcakes per person?
Slide 23 / 130 11 John's car can travel 94.5 miles on 3 gallons of gas. How many miles per gallon can the car travel?
Slide 24 / 130 12 The snake can slither 24 feet in half a day. How many feet can the snake move in an hour?
Slide 25 / 130 13 There are six chaperones at the dance of 100 students. How many students per chaperone are there?
Slide 26 / 130 14 The recipe calls for 6 cups of flour for every four eggs. How many cups of flour are needed for one egg?
Slide 27 / 130 Population Density 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 2009... New Jersey: 1,184 people per square mile Montana: 7 people per square mile California: 237 people per square mile
Slide 28 / 130 15 The population of Newark, NJ is 278,980 people in 24.14 square miles. What is its population density?
Slide 29 / 130 16 The population of Moorestown, NJ is 19,509 people in 15 square miles. What is its population density?
Slide 30 / 130 17 The population of Waco, TX is 124,009 people in 75.8 square miles. What is its population density?
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!
Slide 33 / 130 Identify the conversion factor that results in the desired unit. 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
Slide 34 / 130 Identify the conversion factor that results in the desired unit. 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.
Slide 35 / 130 Identify the conversion factor that results in the desired unit. 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
Slide 37 / 130 18 Write the equivalent rate. 40 mi? mi 1 min 1 h
Slide 38 / 130 19 Write the equivalent rate. 54 inches? inches 1 year 1 month
Slide 39 / 130 20 Write the equivalent rate. 1 day 1week $75? dollars
Slide 40 / 130 21 Write the equivalent rate. 30 sec 1min 425 mi? miles
Slide 41 / 130 22 Write the equivalent rate. 40 feet inches 3 hrs hr Hint: Find the equivalent rate and then determine the unit rate
Slide 42 / 130 23 Write the equivalent rate. 20,000 feet? feet 4 seconds minute Hint: Find the equivalent rate and then determine the unit rate
Slide 43 / 130 24 Write the equivalent rate. 1200 people? people 6 days hr Hint: Find the equivalent rate and then determine the unit rate
Slide 44 / 130 Proportions Return to Table of Contents
Slide 45 / 130 A proportion is an equation that states that two ratios are equivalent. Example: 2 12 3 18 5 15 9 27
Slide 46 / 130 If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Example 1: 2 6 3 x x 3 2 6 3 x Hint: To find the value of x, multiply 3 by 3 also. 2 6 3 9 x 3
Slide 47 / 130 If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Example: 28 7 32 x 4 28 7 32 x Hint: To find the value of x, divide 32 by 4 also. 28 7 32 8 4
Slide 48 / 130 25 Solve the proportion using equivalent ratios 2 8 5 x
Slide 49 / 130 26 Solve the proportion using equivalent ratios 4 x 9 36
Slide 50 / 130 27 Solve the proportion using equivalent ratios 7 35 2 x
Slide 51 / 130 28 Solve the proportion using equivalent ratios x 4 60 12
Slide 52 / 130 29 Solve the proportion using equivalent ratios 3 21 x 28
Slide 53 / 130 In a proportion, the cross products are equal. 5 30 2 12 5 12 2 30 60 60
Slide 54 / 130 Proportions can also be solved using cross products. 4 12 5 x Cross multiply 4x = 5 12 4x = 60 x = 15 Solve for x Example 2 7 x 8 48 Cross multiply 8x = 7 48 8x = 336 x = 42 Solve for x
Slide 55 / 130 30 Use cross products to solve the proportion 9 = x 51 17
Slide 56 / 130 31 Use cross products to solve the proportion x = 56 12 96
Slide 57 / 130 32 Use cross products to solve the proportion 45 = _x 18 6
Slide 58 / 130 33 Use cross products to solve the proportion 2 = _x 15 60
Slide 59 / 130 34 Use cross products to solve the proportion 7 = _3 x 21
Slide 60 / 130 Application problems Return to Table of Contents
Slide 61 / 130 Chocolates at the candy store cost $5.99 per dozen. How much does one candy cost? Round your answer to the nearest cent. Solution: $5.99 1 dozen 1 dozen 12 (Use equivalent rates) $5.99 12 $0.50 per candy
Slide 62 / 130 Example 2: 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 1 570 1x 3 570 x 1,710 books
Slide 63 / 130 Example 3: 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 9 125 9x = 5 125 9x = 625 x = 69.44 70 girls
Slide 64 / 130 35 Cereal costs $3.99 for a one pound box. What is the price per ounce? Round your answer to the nearest penny.
Slide 65 / 130 36 Which is the better buy? Brand A: $2.19 for 12 ounces Brand B: $2.49 for 16 ounces A B Brand A Brand B
Slide 66 / 130 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 67 / 130 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 68 / 130 39 The auditorium can hold 1 person for every 5 square feet. It is 1210 square feet. How many people can the auditorium hold?
Slide 69 / 130 40 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 70 / 130 41 Mary received 4 votes for every vote that Jane received. 1250 people voted. How many votes did Jane receive?
Slide 71 / 130 42 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 72 / 130 43 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. A 76 B 123 C 247 D 496 Question from ADP Algebra I End-of-Course Practice Test
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?
Slide 76 / 130 One way to estimate the number of wolves on a mountain is to use the CAPTURE - RECAPTURE METHOD.
Slide 77 / 130 Suppose this represents all the wolves on the mountain.
Slide 78 / 130 Wildlife biologists first find some wolves and tag them.
Slide 79 / 130 Then they release them back onto the mountain.
Slide 80 / 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 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
Slide 82 / 130 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? 27 7 = f 45 27(45) = 7f 1215 = 7f 173.57 = f There are 174 fish in the river
Slide 83 / 130 A whole group is called a POPULATION. 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
Slide 84 / 130 Example: 860 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 4000 93,100,000 860(93,100,000) = 4000x 80,066,000,000 = 4000x 20,016,500 = x 20,016,500 people watched
Slide 85 / 130 Try This: 315 out of 600 people surveyed voted for Candidate A. How many votes can Candidate A expect in a town with a population of 1500?
Slide 86 / 130 Margin of Error 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
Slide 88 / 130
Slide 89 / 130 44 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 90 / 130 45 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. What is the amount you are going to + by?
Slide 91 / 130 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. What is the lower number in your interval?
Slide 92 / 130 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. What is the upper number in your interval?
Slide 93 / 130 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 94 / 130 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. What is the amount you are going to + by?
Slide 95 / 130 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. What is the lower number in your interval?
Slide 96 / 130 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. What is the upper number in your interval?
Slide 97 / 130 Scale Drawings Return to Table of Contents
Slide 98 / 130 Scale drawings are used to represent objects that are either too large or too small for a life size drawing to be useful. 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.
Slide 99 / 130 A scale is always provided with a scale drawing. 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
Slide 100 / 130 Example: 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.
Slide 101 / 130 Example: 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? 1 100 Write the scale as a ratio 1 x 100 = 2950 100x = 2950 x = 29.5 29.5 inches on the map
Slide 102 / 130 Try This: 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?
Slide 103 / 130 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 104 / 130 53 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 105 / 130 54 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 106 / 130 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 107 / 130 56 Find the length of a 72 inch wide door on a scale drawing with a scale 1 inch : 2 feet.
Slide 108 / 130 57 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 109 / 130 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 110 / 130 59 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. A B C D 4 ft by 9 ft 4 ft by 12 ft 4 ft by 8 ft 4 ft by 10 ft
Slide 111 / 130 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. A B C D 3000 miles 2,727 miles 3,300 miles 1,650 miles
Slide 112 / 130 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. 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
Slide 114 / 130 Two objects are similar if they are the same shape but different sizes. In similar objects: corresponding angles are congruent corresponding sides are proportional
Slide 115 / 130 To check for similarity: Check to see that corresponding angles are congruent Check to see that corresponding sides are proportional (Cross products are equal)
Slide 116 / 130 Example: Is the pair of polygons similar? Explain your answer. 4 yd 3 yd 6 yd 4.5 yd 4 3 = 6 4.5 4(4.5) = 6(3) 18 = 18 YES
Slide 117 / 130 Example: Is the pair of polygons similar? Explain your answer. 8 m 5 m 10 m 13 m 5 8 = 10 13 5(13) = 10(8) 65 = 80 NO
Slide 118 / 130 Example: 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
Slide 119 / 130 Try This: Find the value of y in the pair of similar polygons. 15 in y 7.5 in 5 in
Slide 120 / 130 62 Are the polygons similar? You must be able to justify your answer. Yes No 15 ft 9 ft 21 ft 12 ft
Slide 121 / 130 63 Are the polygons similar? You must be able to justify your answer. Yes No 10 m 8 m 2.5 m 2 m
Slide 122 / 130 64 Are the polygons similar? You must be able to justify your answer. Yes No 15 yd 37.5 yd 15 yd 6 yd
Slide 123 / 130 65 Find the measure of the missing value in the pair of similar polygons. 80 80 y 110 110
Slide 124 / 130 66 Find the measure of the missing value in the pair of similar polygons. 17.5 ft 25 ft 25 ft w 18 ft
Slide 125 / 130 67 Find the measure of the missing value in the pair of similar polygons. x 17 m 4 m 4.25 m
Slide 126 / 130 68 Find the measure of the missing value in the pair of similar polygons. 6 mm y 11 mm 38.5 mm
Slide 127 / 130 69 Find the measure of the missing value in the pair of similar polygons. 63 m 13 m 7 m? 119 m
Slide 128 / 130 70 Find the measure of the missing value in the pair of similar polygons. 7 m 63 m? 13 m 9 m 119 m
Slide 129 / 130 71 Find the measure of the missing value in the pair of similar polygons. 2 mm 5 mm 27.5 mm x
Slide 130 / 130