Copyright 2012 Carlson, O Bryan & Joyner 47

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1 3. Determine if the quantities specified in the following situations are proportional. Justify your response in at least two ways. (Be careful to NOT confuse the quantities with the changes in the quantities.) a. An online video rental company charges $1.20 for each video rented. Determine if the total cost of the videos rented and the number of videos rented are proportional. (Think about how the quantities are related as they change together.) The quantities are proportional and are related by a constant multiple. The total cost of videos rented will always be 1.2 times as large as the number of videos rented. The ratio of the cost to number of videos is always $1.20 to Determine if the quantities specified in the following situations are proportional. Justify your response in at least two ways. (Be careful to NOT confuse the quantities with the changes in the quantities.) b. A Pilates gym charges a $55 registration fee, then $19 for each 50-minute class a person attends. Determine if the total cost of belonging to the gym and the number of classes a person has attended are proportional to one another. The quantities are not proportional. The quantities total cost of belonging to the gym and the number of classes a person has attended do not remain in a constant ratio. 48 1

2 3. Determine if the quantities specified in the following situations are proportional. Justify your response in at least two ways. (Be careful to NOT confuse the quantities with the changes in the quantities.) c. A recipe for homemade brownies calls for cup of cocoa powder for every cup of flour. Determine if the number of cups of cocoa powder is proportional to the number of cups of flour for different sized batches of brownies made. The quantities are proportional. The number of cups of cocoa powder will always be times the number of cups of flour. The ratio of the number of cups of flour to the number of cups of cocoa is always 3: Andy and Tim s grandmother gave them each the same model of remote control car for Christmas. Since their cars go the same speed they meet regularly to run them on the school track. On Monday Tim was a little early and started driving his car before Andy. When Andy arrived he started driving his car. When Tim s car completed 7 laps Andy s car completed 3 laps. a. How many laps had Tim s car completed when Andy s car completed 12 laps? Tim s car had completed 4 more laps, or 16 laps. b. Is the number of laps Tim s car has completed proportional to the number of laps Andy s car has completed as they continue to race their cars around the track? Explain your reasoning. 50 2

3 b. Is the number of laps Tim s car has completed proportional to the number of laps Andy s car has completed as they continue to race their cars around the track? Explain your reasoning. No. The ratio of the number of laps that Tim s car has completed to the number of laps that Andy s car has completed does not remain constant. (We can verify this by picking some points and computing the ratio of the value of the output quantity to the value of the input quantity for several points and show that the ratio s value changes.) Andy and Tim s grandmother gave them each the same model of remote control car for Christmas. Since their cars go the same speed they meet regularly to run them on the school track. On Monday Tim was a little early and started driving his car before Andy. When Andy arrived he started driving his car. When Tim s car completed 7 laps Andy s car completed 3 laps. c. Write a formula to determine the number of laps n completed by Tim s car if you know the number of laps m completed by Andy s car. n = m

4 5. Use the following tables of inputs and outputs to answer the below questions. Table 1 x y Table 2 k r Table 3 m p a. Which table(s) gives values of quantities that could be related proportionally? Explain your reasoning. Table 2. Our reasoning can vary. For example, r is always 2.2 times as large as k Use the following tables of inputs and outputs to answer the below questions. Table 1 x y Table 2 k r Table 3 m p b. For each table that defines a proportional relationship between two quantities, determine a formula to relate the values of the two quantities. r = 2.2k 54 4

5 6. Rebecca has a jar of yellow jellybeans. She adds 58 purple jellybeans to the jar and shakes them up so the yellow and purple jellybeans are mixed well. She then takes out a scoop of 90 jellybeans and finds that exactly 15 of them are purple. She pours the 90 jellybeans back into the jar. a. Estimate the total number of jellybeans in the jar. Explain. The total number of jellybeans is estimated to be 348. This is because the ratio of purple jellybeans to the total number of jellybeans in the jar is 15:90 or 1:6. Since we need to maintain this same ratio, the total number of jellybeans needs to be 6 times the number of purple jellybeans in the jar or (6)(58) = Rebecca has a jar of yellow jellybeans. She adds 58 purple jellybeans to the jar and shakes them up so the yellow and purple jellybeans are mixed well. She then takes out a scoop of 90 jellybeans and finds that exactly 15 of them are purple. She pours the 90 jellybeans back into the jar. b. If Rebecca used a smaller scoop to retrieve jellybeans and finds that exactly 5 of them are purple, how many yellow jellybeans do you estimate are in the scoop? The ratio of the number of purple jellybeans to yellow jellybeans is 15:75 or 1:5. There are 5 times as many yellow jellybeans as there are purple jellybeans, resulting in (5)(5) = 25 yellow jellybeans. 56 5

6 7. Quantity A and Quantity B are proportional and x = the value of Quantity A and y = the value of Quantity B. If we know that 3 units of Quantity A is equal to 2 units of Quantity B, write a formula to relate x and y. Answers can vary. Examples of correct formulas include 57 6