KEY TERMS proportion variable means extremes solve a proportion isolate the variable inverse operations

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1 Tagging Sharks Solving Proportions Using Means 3 an Extremes WARM UP Solve each equation. 1. w x c m 5 10 LEARNING GOALS Rewrite proportions to maintain equality. Represent proportional relationships by equations. Develop strategies to solve proportions. Use proportional relationships to solve multistep problems. KEY TERMS proportion variable means extremes solve a proportion isolate the variable inverse operations You have learne how to write proportions an calculate unknown values through scaling up an scaling own. Is there a more efficient strategy that works for any unknown in any proportion? LESSON 3: Tagging Sharks M1-69

2 Getting Starte Mix-N-Match Recall that a proportion is an equation that states that two ratios are equal. A proportion can be written several ways. Each example shows three proportions using the same four quantities. Example 1 Example 2 Proportion Proportion Proportion In each example, use arrows to show how the numbers were rearrange from the: a. first proportion to the secon proportion. b. first proportion to the thir proportion. M1-70 TOPIC 2: Fractional Rates

3 ACTIVITY 3.1 Maintaining Equality with Proportions Because it is impossible to count each iniviual animal, marine biologists use a metho calle the capture-recapture metho to estimate the population of certain sea creatures. In certain areas of the worl, biologists ranomly catch an tag a given number of sharks. After a perio of time, such as a month, they recapture a secon sample of sharks an count the total number of sharks as well as the number of recapture tagge sharks. Then, the biologists use proportions to estimate the population of sharks living in a certain area. Biologists can set up a proportion to estimate the total number of sharks in an area. Original number of tagge sharks Total number of sharks in an area Number of recapture tagge sharks 5 Number of sharks caught in the secon sample Although capturing the sharks once is necessary for tagging, it is not necessary to recapture the sharks each time. At times, the tags can be observe through binoculars from a boat or at shore. Biologists originally caught an tagge 24 sharks off the coast of Cape Co, Massachusetts, an then release them back into the bay. The next month, they caught 80 sharks with 8 of the sharks alreay tagge. To estimate the shark population off the Cape Co coast, biologists set up the following proportion: 24 tagge sharks 8 5 recapture tagge sharks p total sharks 80 total sharks Notice the variable p in the proportion. In this proportion, let p represent the total shark population off the coast of Cape Co. A variable is a letter or symbol use to represent a number. LESSON 3: Tagging Sharks M1-71

4 1. Write three aitional ifferent proportions you coul use to etermine the total shark population off the coast of Cape Co. 2. Estimate the total shark population using any of the proportions. 3. Di any of the proportions seem more efficient than the other proportions? 4. Willife biologists tag eer in willife refuges. They originally tagge 240 eer an release them back into the refuge. The next month, they observe 180 eer, of which 30 eer were tagge. Approximately how many eer are in the refuge? Write a proportion an show your work to etermine your answer. M1-72 TOPIC 2: Fractional Rates

5 A proportion of the form a b 5 c can be written in many ifferent ways. Another example is b 5 a c or c a 5 b. 5. Show how the variables were rearrange from the proportion in the if statement to each proportion in the then statement to maintain equality. If a b 5 c, then b 5 c a. If b 5 c a, then c a 5 b. 6. Write all the ifferent ways you can rewrite the proportion a b 5 c an maintain equality. ACTIVITY 3.2 Solving Proportions with Means an Extremes The Reay Steay Battery Company tests batteries as they come through the assembly line an then uses a proportion to preict how many of its total prouction might be efective. On Friay, the quality controller teste every tenth battery an foun that of the 320 batteries teste, 8 were efective. If the company shippe a total of 3200 batteries, how many might be efective? A quality control epartment checks the prouct a company creates to ensure that the prouct is not efective. LESSON 3: Tagging Sharks M1-73

6 NOTES Let s analyze a few methos. John Davi 8 efective batteries = efective batteries 320 batteries 3200 batteries = = 80 So, 80 batteries might be efective. Parker efective batteries : 320 batteries efective batteries : 3200 batteries = 80 About 80 batteries will probably be efective How are Parker s an John Davi s methos similar? M1-74 TOPIC 2: Fractional Rates

7 Nora NOTES 8 efective batteries = 320 batteries 1 efective battery 40 batteries = 80 efective batteries 3200 batteries One out of every 40 batteries is efective. So, out of 3200 batteries, 80 batteries coul be efective because = Describe the strategy Nora use. Natalie When I write Nora s ratios using colons like Parker, I notice something about proportions... 8 : 320 = 1 : 40 8 : : 40 1 : 40 = 80 : = the two mile numbers have the same prouct as the two outsie numbers. So, I can solve any proportion by setting these two proucts equal to each other. 3. Verify that Natalie is correct. LESSON 3: Tagging Sharks M1-75

8 4. Try the various proportion-solving methos on these proportions an etermine the unknown value. Explain which metho you use. a. 3 granola bars 420 calories 5 g granola bars 140 calories b. 8 correct : 15 questions 5 24 correct : q questions c. ollars 5 miles 5 $9 7.5 miles Multiplying the means an extremes is like crossmultiplying. The relationship that Natalie notice is between the means an extremes. In a proportion that is written a : b 5 c :, the prouct of the two values in the mile (the means) equals the prouct of the two values on the outsie (the extremes). extremes a b 5 c a : b 5 c : or means extremes means bc 5 a bc 5 a when b Þ 0, Þ 0 To solve a proportion using this metho, first, ientify the means an extremes. Then, set the prouct of the means equal to the prouct of the extremes an solve for the unknown quantity. To solve a proportion means to etermine all the values of the variables that make the proportion true. M1-76 TOPIC 2: Fractional Rates

9 WORKED EXAMPLE You can rewrite a proportion as the prouct of the means equal to the prouct of the extremes. 7 books : 14 ays 5 3 books : 6 ays means extremes 7 books 14 ays 5 3 books 6 ays (14)(3) 5 (7)(6) (14)(3) 5 (7)(6) You can rewrite the prouct of the means an extremes from the worke example as four ifferent equations. Analyze each equation. 3 5 (7)(6) (7)(6) 3 (3)(14) (3)(14) a. Why are these equations all true? Explain your reasoning. b. Compare these equations to the equation showing the prouct of the means equal to the prouct of the extremes. How was the balance of the equation maintaine in each? 6. Why is it important to maintain balance in equations? LESSON 3: Tagging Sharks M1-77

10 NOTES WORKED EXAMPLE In the proportion a b 5 c, you can multiply both sies by b to isolate the variable a. b? a b 5 b? c a 5 bc When you isolate the variable in an equation, you perform an operation, or operations, to get the variable by itself on one sie of the equals sign. Multiplication an ivision are inverse operations. Inverse operations are operations that uno each other. WORKED EXAMPLE Another strategy to isolate the variable a is to multiply the means an extremes an then isolate the variable by performing inverse operations. a b 5 c Step 1: Step 2: Step 3: a 5 bc a 5 bc a 5 bc 7. Describe each step shown. 8. Rewrite the proportion a b 5 c to isolate each of the other variables: b, c, an. Explain the strategies you use to isolate each variable. M1-78 TOPIC 2: Fractional Rates

11 ACTIVITY 3.3 Solving Problems with Proportions Write an solve proportions to solve each problem. 1. An astronaut who weighs 85 kilograms on Earth weighs 14.2 kilograms on the Moon. How much woul a person weigh on the Moon if they weigh 95 kilograms on Earth? Roun your answer to the nearest tenth. 2. Water goes over Niagara Falls at a rate of 180 million cubic feet every 1 2 hour. How much water goes over the Falls in 1 minute? 3. The value of the U.S. ollar in comparison to the value of foreign currency changes aily. Complete the table shown. Roun to the nearest hunreth. Euro U.S. Dollar Do you see how to set up proportions by using two ifferent rows of the table? 4. To make 4.5 cups of fruity granola, the recipe calls for 1.5 cups of raisins, 1 cup of granola, an 2 cups of blueberries. If you want to make 18 cups of fruity granola, how much of each of the ingreients o you nee? LESSON 3: Tagging Sharks M1-79

12 NOTES TALK the TALK Choose Your Own Proportion Aventure Write a problem situation for each proportion. Show the solution n h 1 M1-80 TOPIC 2: Fractional Rates

13 Assignment Write Describe a situation in which you woul use each. 1. variable 2. means an extremes 3. inverse operations 4. isolate the variable Remember To solve a proportion means to etermine all the values of the variables that make the proportion true. You can rewrite a proportion as the prouct of the means an extremes. If a b 5 c, then bc 5 a. Practice Write an solve a proportion to answer each question. 1. Carmen is making a strawberry rink. The recipe calls for 5 parts strawberry juice to 3 parts water. Carmen woul like to make 64 flui ounces of the strawberry rink. How many flui ounces of strawberry juice an water oes Carmen nee? 2. Elena is making a grape rink. The recipe calls for 2 parts grape juice concentrate to 6 parts water. Elena woul like to make 80 flui ounces of the grape rink. How many flui ounces of grape juice concentrate an water oes Elena nee? 3. Jose is making a trail mix. The recipe calls for 3 parts golen raisins to 2 parts cashews. Jose woul like to make 30 cups of trail mix. How many cups of golen raisins an cashews oes Jose nee? 4. Miguel is making a snack mix. The recipe calls for 6 parts of spicy tortilla chips to 3 parts of corn chips. Miguel woul like to make 45 cups of snack mix. How many cups of spicy tortilla chips an corn chips oes Miguel nee? 5. Carla is making a bean sala. The recipe calls for 4 parts green beans to 3 parts yellow wax beans. Carla woul like to make 56 ounces of bean sala. How many ounces of green beans an yellow wax beans oes Carla nee? 6. Shawna is making smoothies. The recipe calls for 2 parts yogurt to 3 parts blueberries. Shawna wants to make 10 cups of smoothie mix. How many cups of yogurt an blueberries oes Shawna nee? Stretch The wor four has 4 letters, so the number-to-letter-count ratio for 4 is 1 : 1, or just 1. Are there any other numbers between 1 an 20 that have equal number-to-letter-count ratios? What are they? LESSON 3: Tagging Sharks M1-81

14 Review 1. The table shows the gallons fille in a pool over time. a. Complete the table. Number of Hours Gallons Fille 500 b. Determine a unit rate for this situation. 2. Pi is the ratio of what two measures of a circle? 3. Which is a better eal a 16-inch iameter pizza for $12.99 or an 8-inch iameter pizza for $6? 4. Evaluate each expression for g a g 1 1 b. 20 g 4 g 1 (8 2 5) M1-82 TOPIC 2: Fractional Rates