2-6 Ratios and Proportions. Determine whether each pair of ratios are equivalent ratios. Write yes or no. SOLUTION: No, the ratios are not equivalent.

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1 Determine whether each pair of ratios are equivalent ratios. Write yes or no No, the ratios are not equivalent Yes, the ratios are equivalent RACE Jennie ran the first 6 miles of a marathon in 58 minutes. If she is able to maintain the same pace, how long will it take her to finish the 26.2 miles? Let t represent the time it will take her to finish 26.2 miles. Write a proportion. No, the ratios are not equivalent. Solve each proportion. If necessary, round to the nearest hundredth. 4. It will take Jennie minutes or 4 hours and 13.3 minutes. 8. MAPS On a map of North Carolina, Raleigh and Asheville are about 8 inches apart. If the scale is 1 inch = 12 miles, how far apart are the cities? 5. esolutions Manual - Powered by Cognero Let d represent the distance between the two cities. Write a proportion. Page 1

2 2-6 Ratios andjennie Proportions It will take minutes or 4 hours and 13.3 minutes. 8. MAPS On a map of North Carolina, Raleigh and Asheville are about 8 inches apart. If the scale is 1 inch = 12 miles, how far apart are the cities? Yes, the ratios are equivalent. 11. Use cross products to check if the ratios are equivalent. Let d represent the distance between the two cities. Write a proportion. No, the ratios are not equivalent. Raleigh and Asheville are about 96 miles apart. 12. Determine whether each pair of ratios are equivalent ratios. Write yes or no. Use cross products to check if the ratios are equivalent. 9. Yes, the ratios are equivalent. No, the ratios are not equivalent Use cross products to check if the ratios are equivalent. Yes, the ratios are equivalent. 11. Use cross products to check if the ratios are equivalent. Yes, the ratios are equivalent. 14. Use cross products to check if the ratios are equivalent. No, the ratios are not equivalent. Page 2

3 Yes, the ratios are equivalent Use cross products to check if the ratios are equivalent. Yes, the ratios are equivalent. 18. Solve each proportion. If necessary, round to the nearest hundredth Page 3

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5 They will wash about 341 cars GEOGRAPHY On a map of Florida, the distance between Jacksonville and Tallahassee is 2.6 centimeters. If 2 centimeters = 120 miles, what is the distance between the two cities? 30. CAR WASH The B Clean Car Wash washed 128 cars in 3 hours. At that rate, how many cars can they wash in 8 hours? Let c represent the number of cars they wash in 8 hours. Write a proportion. Use the given scale to convert the map distance to the real world distance. 32. CCSS PRECISION An artist used interlocking building blocks to build a scale model of Kennedy Space Center, Florida. In the model, 1 inch equals 1.67 feet of an actual space shuttle. The model is inches tall. How tall is the actual space shuttle? Round to the nearest tenth. They will wash about 341 cars. 31. GEOGRAPHY On a map of Florida, the distance between Jacksonville and Tallahassee is 2.6 centimeters. If 2 centimeters = 120 miles, what is the distance between the two cities? Use the scale given to convert the model height to the actual space shuttle height. 33. MENU On Monday, a restaurant made $545 from selling 110 hamburgers. If they sold 53 hamburgers on Tuesday, how much did they make? Let p represent their profit from lunch. Write a proportion. esolutions Manual - Powered by Cognero Page 5 Use the given scale to convert the map distance to the real world distance. Their profit from lunch is about $

6 Use the scale given to convert the model height to the actual space shuttle height. 33. MENU On Monday, a restaurant made $545 from selling 110 hamburgers. If they sold 53 hamburgers on Tuesday, how much did they make? 36. Let p represent their profit from lunch. Write a proportion. 37. Their profit from lunch is about $ Solve each proportion. If necessary, round to the nearest hundredth Page 6 ATHLETES

7 So, 130 students in the ninth grade have braces. 42. PAINT Joel used a half gallon of paint to cover 84 square feet of wall. He has 932 square feet of wall to paint. How many gallons of paint should he purchase? 39. Let p represent the number of gallons of paint needed. Write a proportion. 40. ATHLETES At Piedmont High School, 3 out of every 8 students are athletes. If there are 1280 students at the school, how many are not athletes? It will take about 5.55 gallons of paint, so Paul should purchase 6 gallons of paint. Let s represent the number of students who are athletes. If of the students are athletes, then of 43. MOVIE THEATERS the students are not athletes. Write a proportion. So, 800 students are not athletes. 41. BRACES Two out of five students in the ninth grade have braces. If there are 325 students in the ninth grade, how many have braces? Let s represent the number of students that have braces. Write a proportion. a. Write a ratio of the number of indoor theaters to the total number of theaters for each year. b. Do any two of the ratios you wrote for part a form a proportion? If so, explain the real-world meaning of the proportion. a. For each year, divide the number of indoor theaters by the number total number of theaters. 2003:, 2004:, 2007: So, 130 students in the ninth grade have braces. 42. PAINT Joel used a half gallon of paint to cover 84 square feet- of wall.by HeCognero has 932 square feet of wall esolutions Manual Powered to paint. How many gallons of paint should he purchase?, 2005:, 2008:, 2006:, 2009: b. Compare the ratios. Year Indoor , , ,092 Total 35,995 36,652 37,740 Ratio Page 7

8 Ratios andabout Proportions It will take 5.55 gallons of paint, so Paul should purchase 6 gallons of paint. 38,605 39, No, none of the ratios from part a form a proportion. If two of these ratios formed a proportion, the indoor theaters compared to the total number of theaters would produce equivalent fractions. 44. DIARIES In a survey, 36% of the students said that they kept an electronic diary. There were 900 students who kept an electronic diary. How many students were in the survey? 43. MOVIE THEATERS Let s represents the number of students in the survey. 36% can be written as. Now, write a proportion. a. Write a ratio of the number of indoor theaters to the total number of theaters for each year. b. Do any two of the ratios you wrote for part a form a proportion? If so, explain the real-world meaning of the proportion. a. For each year, divide the number of indoor theaters by the number total number of theaters. 2003:, 2004:, 2007:, 2005:, 2008:, 2006:, 2009: b. Compare the ratios. Year Indoor , , , , , , ,605 Total 35,995 36,652 37,740 38,425 38,794 38,834 39,233 Ratio No, none of the ratios from part a form a proportion. If two of these ratios formed a proportion, the indoor theaters compared to the total number of theaters would produce equivalent fractions. There were 2500 students in the survey. 45. MULTIPLE REPRESENTATIONS In this problem, you will explore how changing the lengths of the sides of a shape by a factor changes the perimeter of that shape. a. GEOMETRIC Draw a square ABCD. Measure and label the sides. Draw a second square MNPQ with sides twice as long as ABCD. Draw a third square FGHJ with sides half as long as ABCD. b. TABULAR Complete the table below using the appropriate measures. c. VERBAL Make a conjecture about the change in the perimeter of a square if the side length is increased or decreased by a factor. a. 44. DIARIES In a survey, 36% of the students said that they kept an electronic diary. There were 900 students who kept an electronic diary. How many students were in the survey? Let s represents the number of students in the survey. 36% can be written as proportion. Page 8. Now, write a b.

9 There were 2500 students in the survey. 45. MULTIPLE REPRESENTATIONS In this problem, you will explore how changing the lengths of the sides of a shape by a factor changes the perimeter of that shape. a. GEOMETRIC Draw a square ABCD. Measure and label the sides. Draw a second square MNPQ with sides twice as long as ABCD. Draw a third square FGHJ with sides half as long as ABCD. b. TABULAR Complete the table below using the appropriate measures. c. VERBAL Make a conjecture about the change in the perimeter of a square if the side length is increased or decreased by a factor. c. If the length of a side is increased by a factor, the perimeter is also increased by that factor. If the length of the sides are decreased by a factor, the perimeter is also decreased by the same factor. 46. CCSS STRUCTURE In 2007, organic farms occupied 2.6 million acres in the United States and produced goods worth about $1.7 billion. Divide one of these numbers by the other and explain the meaning of the result. This the average amount of food produced (in dollar amount) per acre of land used. a acres is the average land area used to produce a dollars worth of goods. 47. REASONING Compare and contrast ratios and rates. Ratios and rates each compare two numbers by using division. However, rates compare two measurements that involve different units of b. measure. An example of a ratio is example of a rate is c. If the length of a side is increased by a factor, the perimeter is also increased by that factor. If the length of the sides are decreased by a factor, the perimeter is also decreased by the same factor. 46. CCSS STRUCTURE In 2007, organic farms occupied 2.6 million acres in the United States and produced goods worth about $1.7 billion. Divide one of these numbers by the other and explain the meaning of the result. and, find the. (Hint: Choose different values of a and b for which the proportions are true and evaluate the expression.) The hint suggests that we choose different values of a and b for which the proportions are true and evaluate the expression This the average amount of food produced (in dollar amount) per acre of land used CHALLENGE If value of. An. How do we go about choosing these numbers? Let s look at the first equation. Page 9 What values of a and b make this equation true? The simplest values to use are a = 4 and b = 2, because 4

10 How do we go about choosing these numbers? Let s 2-6 Ratios Proportions look at and the first equation. What values of a and b make this equation true? The simplest values to use are a = 4 and b = 2, because = 5 and 2 1 = 1, making the numerators and denominators both equal. If the values for a and b did not work for the second equation, then we would have had to find two different values that work in the first equation and try them in the second equation until we found two that worked for both. 49. WRITING IN MATH On a road trip, Marcus reads a highway sign and then looks at his gas gauge. Marcus s gas tank holds 10 gallons and his car gets 32 miles per gallon at his current speed of 65 miles per hour. If he maintains this speed, will he make it to Atlanta without having to stop and get gas? Explain your reasoning. Now that we have two values that work for the first equation, we can test them in the second equation. As it happens, these values work in the second equation as well. First, determine how much gas is left in the tank. If the tank is about gal of gas left. full, he has about Then determine the number of miles he can drive on the gal of gas. At 32 miles per gallon, he will be able to travel 32 Since Atlanta is 200 miles away, he will run out of gas about 20 miles before reaching the city if he doesn t stop to get gas. and So, or 180 miles.. If the values for a and b did not work for the second equation, then we would have had to find two different values that work in the first equation and try them in the second equation until we found two that worked for both. 49. WRITING IN MATH On a road trip, Marcus reads a highway sign and then looks at his gas gauge. 50. WRITING IN MATH Describe how businesses can use ratios. Write about a real-world situation in which a business would use a ratio. For example, a business can use ratios to compare how many of the potential customers in an area use their business. Also, using ratios, a pizza business can find the number of potential customers in their area and compare them to how many they have. They can do the same with their competitors. Marcus s gas tank holds 10 gallons and his car gets 32 miles per gallon at his current speed of 65 miles per hour. If he maintains this speed, will he make it to Atlanta without having to stop and get gas? Explain your reasoning. 51. In the figure, x : y = 2 : 3 and y : z = 3 : 5. If x = 10, find the value of z. First, determine how much gas is left in the tank. If A 15 B 20 C 25 D 30 the tank is about full, he has about Page 10

11 number of potential customers in their area and compare them to how many they have. They can do the same with their competitors. 51. In the figure, x : y = 2 : 3 and y : z = 3 : 5. If x = 10, find the value of z. Choice C is correct. 52. GRIDDED RESPONSE A race car driver records the finishing times for recent practice trials. What is the mean time, in seconds, for the trials? A 15 B 20 C 25 D 30 First, find y. The mean time is 5.02 seconds. 53. GEOMETRY If Now, find z. is equal to, what is z? Choice C is correct. 52. GRIDDED RESPONSE A race car driver records the finishing times for recent practice trials. What is the mean time, in seconds, for the trials? F 240 G 140 H 120 J 70 is equal to because they are vertical angles. The two right angles are equals and is equal to. Then since all three angles in are equal to all three angles in,. Similar triangles have is similar to proportional sides. So, set up a proportion of the sides. The mean time is 5.02 seconds. Page 11

12 The mean time is 5.02 seconds. 53. GEOMETRY If is equal to Choice G is correct., what is z? 54. Which equation below illustrates the Commutative Property? A (3x + 4y) +2z = 3x + (4y + 2z) B 7(x + y) = 7x + 7y C xyz = yxz Dx+0=x Choice B represents the Distributive Property, Choice A shows the Associative Property, and Choice D represents the Additive Identity. The Commutative Property switches the order of multiplying or adding. Therefore, choice C is correct. Solve each equation. 55. F 240 G 140 H 120 J 70 is equal to because they are vertical angles. The two right angles are equals and is equal to. Then since all three angles in are equal to all three angles in,. Similar triangles have is similar to proportional sides. So, set up a proportion of the means the distance between x and 5 is 8. Since distance cannot be negative, the solution is the empty set. 56. Case 1: Case 2: sides. The solution set is ( 7, 11). 57. Choice G is correct. 54. Which equation below illustrates the Commutative Property? A (3x + 4y) +2z = 3x + (4y + 2z) B 7(x + y) = 7x + 7y C xyz = yxz Dx+0=x Choice B represents the Distributive Property, Choice A shows the Associative Property, and Choice D represents Additive Identity. The esolutions Manual - Powered bythe Cognero Commutative Property switches the order of multiplying or adding. Therefore, choice C is correct. Case 1: Case 2: Page 12

13 Case 2: The solution set is ( 7, 11). The solution set is. 59. HEALTH When exercising, a person s pulse rate should not exceed a certain limit. This maximum rate is represented by the expression 0.8(220 a), where a is age in years. Find the age of a person whose maximum pulse rate is Case 1: Case 2: A person whose maximum pulse rate is 152 is 30 years old. Solve each equation. Check your solution = 4a 5 The solution set is {10, 7}. 58. Case 1: Check: Case 2: 61. 7g 14 = 63 The solution set is. 59. HEALTH When exercising, a person s pulse rate should not exceed a certain limit. This maximum rate is represented by the expression 0.8(220 a), where a is age in years. Find the age of a person whose maximum pulse rate is 152. Check: Page

14 61. 7g 14 = Check: Check: 62. ABC if each small triangle has a base of 5.2 inches and a height of 4.5 inches. 64. GEOMETRY Find the area of Check: The area of one small triangle is: 63. There are 4 small triangles. Multiply the area of the small triangle by 4 to determine the area of the large triangle. So, the area of is or 46.8 square inches. Evaluate each expression. 65. Check: Page

15 There are 4 small triangles. Multiply the area of the small triangle by 4 to determine the area of the large 2-6 Ratios triangle.and So,Proportions the area of is or 46.8 square inches. Evaluate each expression Solve each equation p = h = y = m = Page 15

Grade 8, Unit 3 Practice Problems - Open Up Resources

Grade 8, Unit 3 Practice Problems - Open Up Resources Grade 8, - Open Up Resources Lesson 1 Priya jogs at a constant speed. The relationship between her distance and time is shown on the graph. Diego bikes at a constant speed twice as fast as Priya. Sketch