Algebra 1-8 Study Guide: Rates, Ratios, and Proportions (pp 62-64) Page 1! of 11! 1. 6x = 36 2.! 3. 5m = 18

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1 Page 1! of 11! Attendance Problems: Solve each equation. x 1. 6x = 36 2.! 3. 5m = 18 4 = 12 r 6.! 7. 8y = = 3 Multiply.! 8.! 8 7 $ 9.! " # 8% & 12! 5 $ " # 6 % & Vocabulary ratio proportion rate cross products scale scale drawing unit rate scale model conversion factor dimensional analysis I can write and use ratios, rates, and unit rates. I can write and solve proportions. Common Core CC.9-12.N.Q.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. * CC.9-12.A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Question: What did the circle see when sailing on the ocean? Answer: Pi rates.

2 ! Algebra 1-8 Study Guide: Rates, Ratios, and Proportions (pp 62-64) Page 2! of 11! 10. What is a ratio? 11. What is a proportion? Reading Math Read the proportion as 1 is to 15 as x is to 675. Video Example 1: The ratio of faculty member to students at a college is 1:12. There are 1440 students. How many faculty members are there? 1 Using Ratios The ratio of faculty members to students at a college is 1:15. There are 675 students. How many faculty members are there? faculty _ students _ 1 _ 15 1_ 15 = x_ ( _ 675) x = 675 _ ( 15) 1 x = 45 There are 45 faculty members. Write a ratio comparing faculty to students. Write a proportion. Let x be the number of faculty members. Since x is divided by 675, multiply both sides of the equation by 675.

3 Page 3! of 11! Example 1: The ratio of the number of bones in a human s ears to the number of bones in the skull is 3:11. There are 22 bones in the skull. How many bones are in the ears? 12. Guided Practice: The ratio of games won to games lost for a baseball team is 3:2. The team has won 18 games. How many games did the team lose? 13. What is a rate? 14. How is a unit rate different than a rate? Video Example 2: The price for 8.5 pounds of compost starter is $15. Find the unit rate. 2 Finding Unit Rates Takeru Kobayashi of Japan ate 53.5 hot dogs in 12 minutes to win a contest. Find the unit rate in hot dogs per minute. Round to the nearest hundredth. _ = _ x x Write a proportion to find an equivalent ratio with a second quantity of 1. Divide on the left side to find x. The unit rate is approximately 4.46 hot dogs per minute.

4 Page 4! of 11! Example 2: Raulf Laue of Germany flipped a pancake 416 times in 120 seconds to set the world record. Find the unit rate. Round your answer to the nearest hundredth. 15. Guided Practice: Dylan earns $52.50 in 7 hours. Find the unit rate. 16. What is dimensional analysis? Video Example 3. A. As you go deeper underground, the earths temperature increases. In some places, it may increase by! 20 C per kilometer. What is this rat in degrees per meter? B. The record running speed of a giraffe is 34.7 miles per hour. What is the speed in feet per minute?

5 Page 5! of 11! 3 Using Dimensional Analysis A A large adult male human has about 12 pints of blood. Use dimensional analysis to convert this quantity to gallons. Step 1 Convert pints to quarts. 12 pt _ 1 qt Multiply by a conversion factor whose first 2 pt quantity is quarts and whose second quantity is pints. 6 qt 12 pints is 6 quarts. Step 2 Convert quarts to gallons. 6 qt _ 1 gal 4 qt 6_ 4 gal = 1 _ 1 2 gal Multiply by a conversion factor whose first quantity is gallons and whose second quantity is quarts. A large adult male human has about 1 1 gallons of blood. 2 B The dwarf sea horse Hippocampus zosterae swims at a rate of feet per hour. Use dimensional analysis to convert this speed to inches per minute. 12 in. Use the conversion factor to convert feet to inches, and use the 1 ft 1 h conversion factor to convert hours to minutes. 60 min _ ft _ 12 in. _ 1 h = _ in. 1 h 1 ft 60 min 1 min The speed is inches per minute. Check that the answer is reasonable. The answer is about 10 in./min. There are 60 min in 1 h, so 10 in./min is 60 (10) = 600 in./h. There are 12 in. in 1 ft, so 600 in./h is = 50 ft/h. This is close to the rate given in the problem, ft/h. "Joy is wealth and love is the legal tender of the soul." -- Robert G. Ingersoll

6 Page 6! of 11! Example 3. A. A fast sprinter can run 100 yards in approximately 10 seconds. Use dimensional analysis to convert 100 yards to miles. Round to the nearest hundredth. (Hint: There are 1760 yards in a mile.) Helpful Hint In Additional Example 3A, yd appears to divide out, leaving mi, as the unit. Use this strategy of dividing out units when using dimensional analysis. B. A cheetah can run at a rate of 60 miles per hour in short bursts. What is this speed in feet per minute? 17. Guided Practice: A cyclist travels 56 miles in 4 hours. Use dimensional analysis to convert the cyclist s speed to feet per second? Round your answer to the nearest tenth, and show that your answer is reasonable. 1-8 Rates, Ratios, and Proportions (p 66)

7 Page 7! of 11! Cross Products Property WORDS NUMBERS ALGEBRA In a proportion, cross products are equal. 2 6 = 3 4 If and b 0 and d 0 then ad = bc. Video Example 4. Solve each proportion 2 A. B. 7 = 5 4 w x + 2 = 1 3 Example 4. 3 A.! B.! 9 = 5 6 m y 3 = 2 7

8 Page 8! of 11! 4 Solving Proportions Solve each proportion. A 5_ 9 = _ w 3 5_ 9 = _ 3 w 5 (w) = 9 (3) Use cross products. 5w = 27 5w Divide both sides 5 = 27 5 by 5. w = _ 27 5 B 8_ x + 10 = _ _ x + 10 = _ (12) = 1 (x + 10) Use cross products. 96 = x = x - 10 Subtract 10 from both sides. Guided Practice: Solve _ each proportion ! 19.! 2 = y g + 3 = _ 20. What is a scale?

9 Page 9! of 11! Video Example 5. A. On the map, the distance from Santa Cruz to Santa Maria is about 7.8 in. What is the approximate actual distance? B. The actual distance between Fresno and Bakersfield is 110 miles. What is the distance between these two location on the map?

10 Page 10! of 11! 5 Scale Drawings and Scale Models A On the map, the distance from Chicago to Evanston is in. What is the actual distance? _ map actual 1_ _ 18 = _ x 1 in. _ 18 mi Write the scale as a fraction. Let x be the actual distance. x 1 = 18 (0.625) Use cross products x = to solve. The actual distance is mi. Waukegan North Chicago Highland Park Evanston B The actual distance between North Chicago and Waukegan is 4 mi. What is this distance on the map? Round to the nearest tenth. map _ actual 1_ 18 = _ x 4 _ 4 = 18x 4_ 18 = 18x x _ 1 in. _ 18 mi Write the scale as a fraction. 1 in:18 mi Let x be the distance on the map. The distance on the map is about 0.2 in. Use cross products to solve the proportion. Chicago Since x is multiplied by 18, divide both sides by 18 to undo the multiplication. Round to the nearest tenth.

11 Page! 11 of 11! Example 5. A contractor has a blueprint for a house drawn to the scale 1 in: 3 ft A. A wall on the blueprint is 6.5 inches long. How long is the actual wall? B. One wall of the house will be 12 feet long when it is built. How long is the wall on the blueprint? 21. Guided Practice. A scale model of a human heart is 16 ft. long. The scale is 32:1. How many inches long is the actual heart it represents? 1-8 Rates, Ratios, and Proportions (p 66) 20-25, 33, 35, 37-43, 54,