b. $52.50; Sample explanation: $63 120% 100% 11. (See Figure 1) 12. (See Figure 2) Selling Price

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1 Applications $6.00 = $ $6.80 = $.77 (rounded value). 0.0 $.90 = $1.1 (rounded value) $49.99 = $10.00 (rounded value) $9.9 = $.40 (rounded value) 6. All five strategies are correct. Students opinions as to which strategy makes the most sense will vary. For eample, strategies (C) and (D) describe simple patterns, but they cannot be generalized as easily as strategies (A), (B), and (E). 7. a. $60; Sample eplanation: $6 = where 10% and $6 10% 100% represent the total amount of money the group can spend, including ta. and 100% represent the cost of the before ta. = $60 b. $.0; Sample eplanation: $6 = 10% 100% where $6 and 10% represent the total amount of money the group can spend, including ta and tip. Since ta (%) and tip (1%) are both calculated from the cost of the, 100% + % + 1% = 10% the total amount spent. and 100% represent the total cost of the before ta and tip. = $.0 8. over $.00; % of $40.00 is $.00, so % of $4.00 would be over $ under $6.00; 10% of $60.00 is $6.00, so 9% of $9.99 would be less than $ over $1.00; % of $00.00 is $1.00, so.% of $09.9 would be over $ (See Figure 1) 1. (See Figure ) Figure 1 Costs and Revenue for Roberto's Sales Buying Markup (80% of buying price) Selling Commission (% of markup) Profit (money the shop makes on the sale) $100 $80 $180 $0 $60 $10 $8 $18 $ $6 $ $44 $99 $11 $ $1 $100 $ $ $7 Figure Costs and Revenue for Linda's Sales Buying Markup (80% of buying price) Selling Commission (% of markup) Profit (money the shop makes on the sale) $60 $48 $108 $1 $6 $140 $11 $ $8 $84 $7 $7.60 $19.60 $14.40 $4.0 $90 $7 $18 $18 $4 $N $0.8N, or $ 4 N $1.8N, or $ 9 N $0.N, or $ N $0.6N, or $ N Comparing and Scaling 1 Investigation

2 1. 1. represents the number of gallons in 6 cups; = 1 4 gallons.. represents the number of fluid ounces in. 1 1 cups; = 100 fluid ounces. 16 ounces 1 pound = ; = 168 ounces 101 pounds 4. 1 gallon 16 cups gallons = ; = 880 cups 14. a. markup = 0.8 buying; buying = markup 1. b. selling = buying 0.8. = buying 1.8; buying = selling 9 c. commission = 0. markup; markup = commission 4 d. profit = commission ; commission = profit minutes minutes 17.,840 Calories grams of fat 19. For all of these problems, students should set up some sort of an equation and solve, perhaps using a proportion or some other strategy. a. 10 minutes b. 40 minutes c. 64 minutes d. 880 seconds e. 16 minutes 0. represents the number of ounces in 1 pounds; = 6 ounces.. 1 kilogram 60 kilograms ; = 1 pounds. pounds 6. a. Both Alicia s and Brandon s methods are correct. In Alicia s method, simplifying one side allows you to solve the problem by undoing the division on the left side. Brandon s method works because he simply scales each of the values by 100, which does not change the multiplicative relationship between quantities in the proportion. Note: There is nothing special about 100. Any nonzero quantity will work the same way. Charlene s method does not work because the proportion has a multiplicative, not additive, relationship. b. Answers will vary. 7. unit rate = 8. unit rate = 4 9. unit rate = 0. unit rate = % high-fiber to 0%, high-protein. % high-fiber to 7%, high-protein. 90% high-fiber to 10%, high-protein 4. a. 6; 4 1. = 6 Note: The unit rate of high-protein to high-fiber is 1.. b. 16; 4 = 16 Note: The unit rate of high-protein to high-fiber is. Comparing and Scaling Investigation

3 Connections. The commissions are the same. Suppose the starting price is P. Then the markup in the first situation will be 0.P. The salesperson gets 10, of the markup, so the commission will be 0.10(0.P), or 0.0P. In the second situation, the markup is 0.10P. The commission is 0.(0.10P), or 0.0P. As long as the starting prices are the same, the commissions will be the same. 6. a $4.90 b $67.0 c $99.99 d $ $1; Since this situation involves a discount, you need to subtract %. $1 (0.0 $1) = 0.9 $1 8. a. No, Bill s method will not work. Students may provide an eample, such as a $100 item. The discount on a $100 item would be $6. The ta would then be calculated from the $94 discounted price, not on the original $100. So, the final sale price would be $ b. It does not matter which is applied first, the discount or the ta. Take an item with a starting price of P. If you take the discount before the ta, the equation is 1.06 (0.94P) = P. If you calculate the ta and then the discount, the equation is 0.94 (1.06P) = P. This is because the epression is the product of three values. The values can commute. 9. B; 1 1 =,and 1 < 1 1, so the 4 quotient would be between 1 and. 40. H; 10 =, and 1 7 <, so the quotient 8 would be slightly greater than. 41. B; = 4, and 9 < 6, so the quotient 10 would be between 1 and J; 14 1 = 14, and 14 > 14; 8 < 1, so 7 10 the quotient would be greater than A; = 6 < 7, so the quotient would be less than G; Common denominators yield the eact quotient 19. You can estimate by noticing 1 that 19 0 is slightly less than 1, and 6 10 is slightly greater than 1. Thus, the answer should be greater than 1 but certainly less than. 4. a. Using the diagram, if you divide 4 you would get 1 minutes, or 1 4 hour by, 1 4 miles = 9 4 miles. Dividing 9 miles by 4 is 4 mile. Felipe walked 4 mile. b. You can assume Felipe walks at a constant rate. Using the answer from part (a), he would walk 4 4 miles, in 1 hour. c. Since Felipe is walking at a rate of miles, or miles per hour, it would take him 90 minutes to walk 4 1 miles. You can also think of 4 1 as twice 1 4. Doubling both the miles walked and the time it took Felipe to walk would result in 1 1 hours. d. 1 hour and minutes or hours; By partitioning the segment between 46. = 6 and 4 miles into three pieces, the hours segment between 1 and 1 1 4, or 1, is also partitioned into three equal 1 pieces. 47. = = = 0. a. There are liters. b. In. liters there are =.8 quarts. c. From the unit rates, Q = 1.06L and L = 0.94Q. Comparing and Scaling Investigation

4 1. a. 1 : 7, or : 1 b. P = F or P = F c. 4; Substitute 1 for P, and solve for F. d. 1 scoop e. Orangutan Food Mi. a. b , or 7 =. Baby Gorilla Food Mi Scoops of High-Protein Food Scoops of High-Fiber Food 6 7 c. Baby Gorilla Food Mi 1 7 d. P = 7 F or F = 7 P. a. about 4 scoops b. scoops of high-protein : 6 scoops of high-fiber c. P = 6 F d. 4 scoops Comparing and Scaling Investigation 4

5 4. a. Answers will vary. Sample answer: The camps were relatively close in terms of the difference between boys and girls. 4 more girls attended Camp Green than Camp Blue. more boys attended Camp Green than Camp Blue. There were 0 more boys than girls at Camp Green and 40 more boys than girls at Camp Blue. Since 4 more girls attended Camp Green than Camp Blue, Camp Green must appeal more to girls. b. Answers will vary. Sample answer: The total number of campers at Camp Green is = 00. The fraction of boys at Camp Green is then 1 00 = 8. The fraction for girls at Camp Green is 7 00 =.The total for Camp Blue is = 100. The fraction of boys at Camp Blue is = 7 and the fraction 10 of girls at Camp Blue is = 10. You can then compare fractions with like denominators, for eample, compare 8 = for boys at Camp Green to 40 8 = 1 for girls at Camp Green and for Camp Blue become 8 boys and girls for Camp Blue. Since 1 of the campers at Camp Green were girls, and only 1 of the campers at Camp 40 Blue were girls, then Camp Green must appeal more to girls than Camp Blue. c. Answers will vary. Sample answer: 6.% of campers at Camp Green were boys, and 70% of campers at Camp Blue were boys. 7.% of campers at Camp Green were girls and 0% at Camp Blue were girls. The percentage of campers who were girls is greater for Camp Green than Camp Blue, so Camp Green must appeal more to girls. d. Answers will vary. Sample answer: The ratio of to describes boys to girls at Camp Green. A ratio of 7 to describes boys to girls at Camp Blue. The ratio of boys to girls is greater at Camp Blue than Camp Green, so Camp Green must appeal more to girls than Camp Blue.. a. football; The ratio of boys to girls is 6 : 1, the most etreme ratio of all the sports. b. soccer c. i. rounded to the nearest whole number, basketball = 89, football = 67, soccer = 94 ii. rounded to the nearest whole number, basketball = 4, football = 1, soccer = 180 Comparing and Scaling Investigation