0DNLQJ3XQFK Ratios, Rates, and Mixture Problems Learning Goals In this lesson, you will: f Use ratios to make comparisons. f Use rates and proportions to solve mixture problems. D o you like smoothies? Perhaps one of the best things about smoothies is that you can make one with just about any ingredients. Just throw them in the blender and turn it on! Smoothies can be very healthy too. Try this healthy smoothie recipe sometime. ● 1 banana ● 1 cup of vanilla yogurt ● 1 cup of grapes ● 1 __ ● 2 cups of spinach leaves 2 of an apple If this recipe serves 3 people, how much of each ingredient would you need to © 2011 Carnegie Learning make smoothies your whole class? H 1.2 Ratios, Rates, and Mixture Problems 15 Problem 1 Four recipes for grapefruit punch are given. Students write ratios that compare the number of parts of grapefruit juice to the total number of parts in each of the four given recipes. Next, they will write ratios that compare the number of parts of lemon-lime soda to the total number of parts in each recipe. Problem 1 Each year, your class presents its mathematics portfolio to parents and community members. This year, your homeroom is in charge of the refreshments for the reception that follows the presentations. Four students in the class give their recipes for punch. The class wants to analyze the recipes to determine which will make the punch with the strongest grapefruit flavor, and which will make the strongest lemon-lime soda flavor. The recipes are shown. Bobbi’s Recipe Adam’s Recipe Grouping t Ask a student to read the 4 parts lemon-lime soda 3 parts lemon-lime soda 8 parts grapefruit juice 5 parts grapefruit juice Zeb’s Recipe Carlos’s Recipe information and recipes aloud. Complete Question 1 as a class. t May the Best Recipe Win 2 parts lemon-lime soda 1 part lemon-lime soda 3 parts grapefruit juice 4 parts grapefruit juice 1. How many total parts are in each person’s recipe? Have students complete Questions 2 through 5 with a partner. Then share the responses as a class. Adam’s recipe: 12 total parts; Bobbi’s recipe: 8 total parts; Carlos’s recipe: 5 total parts; Zeb’s recipe: 5 total parts 2. For each recipe, write a ratio that compares the number of parts of grapefruit juice to the total number of parts in each recipe. If possible, simplify each rate. 8 parts grapefruit juice 2 parts grapefruit juice Adam’s recipe: _____________________ 5 _____________________ 12 total parts 3 total parts 5 parts grapefruit juice Bobbi’s recipe: _____________________ 8 total parts 3 parts grapefruit juice Carlos’s recipe: _____________________ 5 total parts H 16 Chapter 1 Ratios and Rates © 2011 Carnegie Learning 4 parts grapefruit juice Zeb’s recipe: _____________________ 5 total parts Share Phase, Questions 2 through 5 t Who’s recipe contains the 3. Which recipe will make the punch with the strongest grapefruit taste? Explain how most number of parts? you determined your answer. t Who’s recipe contains the least number of parts? t Can any of the ratios be reduced? Which ones? 96 80 ; __ 5 5 ____ 75 ; __ 3 5 ____ 4 5 ____ 72 ; __ 2 5 ____ Sample Answer: __ 3 120 8 120 5 120 5 120 5, and __ 3 , Zeb’s recipe will have the strongest taste 4 is greater than __ 2, __ Because __ 5 5 3 8 of grapefruit flavor. t Are these ratios considered part-to-part or part-to-whole ratios? Explain. t Which recipe will have the weakest taste of grapefruit? Explain. t Which recipe will have the weakest taste of lemon-lime soda? Explain. First, I had to scale up all the rates to rates with a common denominator. Then, I was able to determine which recipe would have the strongest grapefruit flavor. 4. For each recipe, write a rate that compares the number of parts of lemon-lime soda to the total number of parts in each recipe. If possible, simplify each rate. 4 parts lemon-lime soda 1 part lemon-lime soda Adam’s recipe: _______________________ 5 ______________________ 12 total parts 3 total parts 3 parts lemon-lime soda Bobbi’s recipe: _______________________ 8 total parts 2 parts lemon-lime soda Carlos’s recipe: _______________________ 5 total parts 1 part lemon-lime soda Zeb’s recipe: ______________________ 5 total parts 5. Which recipe will make the punch with the strongest lemon-lime soda flavor? Explain how you determined your answer. 48 ; __ 40 ; __ 3 5 ____ 45 ; __ 24 2 5 ____ 1 5 ____ 1 5 ____ Sample Answer: __ 3 120 8 120 5 120 5 120 3, and __ 1 , Carlos’s recipe will have the strongest taste 1, __ 2 is greater than __ Because __ 5 5 3 8 of lemon-lime soda flavor. First, I had to scale up all the rates to rates with a common denominator. Then, I was able to determine which recipe would have the strongest lemon-lime © 2011 Carnegie Learning soda flavor. H 1.2 Ratios, Rates, and Mixture Problems 17 Problem 2 Each glass of punch contains 6 fluid ounces. One hundred sixty people will attend the reception. Students use proportions to determine unit rates and numbers of fluid ounces of punch are needed for each part if they were to use each recipe for punch in Problem 1. They will then organize their data in a table and answer questions related to the data. Problem 2 Making the Refreshments 1. You are borrowing glasses from the cafeteria to serve the punch. Each glass holds 6 fluid ounces of punch. Your class expects that 70 students and 90 parents and community members will attend the reception. You decide to make enough punch so that every person who attends can have one glass of punch. How many fluid ounces of punch will you need for the reception? Number of people attending: 70 1 90 5 160 Number of fluid ounces: 160(6) 5 960 I will need 960 fluid ounces of punch for the reception. Previously, you wrote rates to compare parts of each ingredient to total parts of all the ingredients. Recall that a rate is a ratio in which the units of the parts or the whole being Grouping compared are different. Have students complete Questions 1 through 7 with a partner. Then have students share their responses as a class. 2. Determine the unit rate for the fluid ounces of punch there would be in one part of the recipe if your class uses Adam’s recipe. 80 fluid ounces 960 fluid ounces 5 _______________ ________________ 1 part 12 parts There will be 80 fluid ounces of punch for each part of the recipe. 3. How many fluid ounces of lemon-lime soda and grapefruit juice are needed to make enough punch if your class uses Adam’s recipe? Show all your work. Number of fluid ounces of lemon-lime soda: x fluid ounces 80 fluid ounces 5 _____________ _______________ 1 part 4 parts (80)(4) 5 320 x 5 320 Number of fluid ounces of grapefruit juice: x fluid ounces 80 fluid ounces 5 _____________ _______________ 1 part 8 parts (80)(8) 5 640 x 5 640 I will need 320 ounces of lemon-lime soda and 640 ounces of grapefruit juice to H 18 Chapter 1 Ratios and Rates © 2011 Carnegie Learning make enough punch if my class uses Adam’s recipes. Share Phase, Questions 1 through 5 t How did you determine the number of fluid ounces of punch needed? t t t t 4. How many fluid ounces of lemon-lime soda and grapefruit juice are needed to make enough punch if your class uses Bobbi’s recipe? Show all your work. Number of fluid ounces in one part of recipe: 120 fluid ounces 960 fluid ounces 5 ________________ ________________ 1 part 8 parts How did you determine the unit rate for the number of fluid ounces of punch there would be in one part of the recipe? Number of fluid ounces of lemon-lime soda: x fluid ounces 120 fluid ounces 5 _____________ ________________ 1 part 3 parts What is 960 divided by 12? Number of fluid ounces of grapefruit juice: Are you scaling up or scaling down to solve the proportions in this problem scenario? How did you determine the amount of lemon-lime soda needed in each recipe? (120)(3) 5 360 x 5 360 x fluid ounces 120 fluid ounces 5 _____________ ________________ 1 part 5 parts (120)(5) 5 600 x 5 600 I will need 360 ounces of lemon-lime soda and 600 ounces of grapefruit juice to make enough punch if my class uses Bobbi’s recipe. 5. How many fluid ounces of lemon-lime soda and grapefruit juice are needed to make enough punch if your class uses Carlos’s recipe? Show all your work. Number of fluid ounces in one part of recipe: 192 fluid ounces 960 fluid ounces 5 ________________ ________________ 5 parts 1 part Number of fluid ounces of lemon-lime soda: x fluid ounces 192 fluid ounces 5 _____________ ________________ 1 part 2 parts (192)(2) 5 384 x 5 384 Number of fluid ounces of grapefruit juice: x fluid ounces 192 fluid ounces 5 _____________ ________________ 1 part 3 parts (192)(3) 5 576 x 5 576 I will need 384 ounces of lemon-lime soda and 576 ounces of grapefruit juice to © 2011 Carnegie Learning make enough punch if my class uses Carlos’s recipe. H 1.2 Ratios, Rates, and Mixture Problems 19 Share Phase, Questions 6 and 7 t How did you determine the 6. How many fluid ounces of lemon-lime soda and grapefruit juice are needed to make enough punch for the reception if your class uses Zeb’s recipe? Show all your work. amount of grapefruit juice needed in each recipe? Number of fluid ounces in one part of recipe: 192 fluid ounces 960 fluid ounces 5 ________________ ________________ t Using the data in the table, 5 parts 1 part Number of fluid ounces of lemon-lime soda: which recipe has the weakest taste of grapefruit? x fluid ounces 192 fluid ounces 5 _____________ ________________ 1 part t Using the data in the table, 1 part 192(1) 5 192 x 5 192 which recipe has the weakest taste of lemon-lime? Number of fluid ounces of grapefruit juice: x fluid ounces 192 fluid ounces 5 _____________ ________________ 1 part 4 parts 192(4) 5 768 x 5 768 I will need 192 ounces of lemon-lime soda and 768 ounces of grapefruit juice to make enough punch if my class uses Zeb’s recipe. H 20 Chapter 1 Ratios and Rates Amount of Lemon-Lime Soda (fluid ounces) Amount of Grapefruit Juice (fluid ounces) Total Amount of Punch (fluid ounces) Adam’s recipe 320 640 960 Bobbi’s recipe 360 600 960 Carlos’s recipe 384 576 960 Zeb’s recipe 192 768 960 © 2011 Carnegie Learning 7. Complete the table with the calculations you determined for each person’s recipe. Grouping Have students complete Questions 8 through 11 with a partner. Then share the responses as a class. Share Phase, Questions 8 through 11 t If the recipe has the weakest taste of grapefruit, does that mean it has the strongest taste of lemon-lime soda? t t t What would the punch taste like if there were equal parts of grapefruit juice and lemonlime soda used in the recipe? If you put more grapefruit juice in the punch, does that result in a stronger grapefruit taste? If you put more lemon-lime soda in the punch, does that result in a stronger lemonlime soda taste? grapefruit flavor? How does the table confirm your choice? Zeb’s recipe will have the strongest grapefruit flavor. His recipe uses the most grapefruit juice, 768 fluid ounces, out of the total 960 fluid ounces. 9. In Problem 1, Question 5 you determined which recipe would have the strongest lemon-lime soda flavor? How does the table confirm your choice? Carlos’s recipe will have the strongest lemon-lime soda flavor. His recipe has the most lemon-lime soda, 384 fluid ounces, out of the total 960 fluid ounces. 10. If you would use 8-ounce glasses for the reception rather than 6-ounce glasses, how would that affect the amount of punch you would need to make? There are 160 people attending. Number of total fluid ounces: (160)(8) 5 1280 1280 2 960 5 320 To use 8-ounce glasses rather than 6-ounce glasses would require 320 more fluid 1 more punch. ounces of punch, or about __ 3 11. Will the ratio of the parts for any of the recipes change by putting more punch in each glass? Explain your reasoning. The ratio of parts of any recipe will remain the same. © 2011 Carnegie Learning t If the recipe has the weakest taste of lemon-lime soda, does that mean it has the strongest taste of grapefruit juice? 8. In Problem 1, Question 3 you determined which recipe would have the strongest H 1.2 Ratios, Rates, and Mixture Problems 21 Talk the Talk Students explain how using ratios and rates helped them solve the problems in this lesson. Talk the Talk 1. Explain how ratios and rates helped you solve the problems in this lesson. The ratios in Problem 1, Questions 1 and 2, indicated which punch tasted more like grapefruit and which punch tasted more like lemon-lime soda. Rates were used to Grouping determine how many fluid ounces of lemon-lime soda and grapefruit juice were needed to make each recipe. Have students complete Question 1 with a partner. Then share the responses as a class. H 22 Chapter 1 Ratios and Rates © 2011 Carnegie Learning Be prepared to share your solutions and methods.

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