Copyright © Ed2Net Learning, Inc. 1 Algebra I Rates, Ratios, and Proportions

Copyright © Ed2Net Learning, Inc. 2 Warm Up 1) 2c - 5 = c + 4 2) 8r + 4 = r 3) 2x -1 = x ) y = 0.7y - 12 Factor each trinomial by guess and check:

Copyright © Ed2Net Learning, Inc. 3 Rates, Ratios, and Proportions A ratio is a comparison of two quantities by division. The ratio of a to b can be written a:b or a, b where b ≠ 0. Ratios that name the same comparison are said to be equivalent. A statement that two ratios are equivalent, such as 1 = 2, is called a proportion Read the proportion 1 = x “1 is to 15 as x is to 675.”

Copyright © Ed2Net Learning, Inc. 4 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 = 1 Students 15 1 = x = x 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.

Copyright © Ed2Net Learning, Inc. 5 Now you try! 1) The ratio of games won to games lost for a baseball team is 3 : 2. The team won 18 games. How many games did the team lose?

Copyright © Ed2Net Learning, Inc. 6 A rate is a ratio of two quantities with different units, such as or 34mi, 2gal Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as 34mi, 2gal or 17 mi/gal. You can convert any rate to a unit rate.

Copyright © Ed2Net Learning, Inc. 7 Garry ate 53.5 hot dogs in 12 minutes to win a contest. Find the unit rate. Round your answer to the nearest hundredth. Finding Unit Rates 4.46 ≈ x Write a proportion to find an equivalent ratio with a second quantity of 1. Divide on the left side to find x = x 12 1 The unit rate is approximately 4.46 hot dogs per minute.

Copyright © Ed2Net Learning, Inc. 8 Now you try! 1) Cory earns $52.50 in 7 hours. Find the unit rate.

Copyright © Ed2Net Learning, Inc. 9 A rate such as 12in., 1 ft in which the two quantities are equal but use different units, is called a conversion factor. To convert a rate from one set of units to another, multiply by a conversion factor. Conversion factor

Copyright © Ed2Net Learning, Inc. 10 Conversion factor A) The earth’s temperature increases, as you go deeper underground. In some places, it may increase by 25°C per kilometer. What is this rate in degrees per meter? To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. 25°C × 1km 1km 1000m The rate is 0.025°C per meter °C 1m

Copyright © Ed2Net Learning, Inc. 11 B) The dwarf sea horse Hippocampus zosterae swims at a rate of feet per hour. What is this speed in inches per minute? To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity ft × 12 in 1h 1ft The speed is inches per hour in. 1h Step1: Convert the speed to inches per hour. Next page 

Copyright © Ed2Net Learning, Inc. 12 To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity in × 1 h 1h 60 min The speed is inches per minute in 1 min Step2: Convert this speed to 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 600 = 50 ft/h. This is close to the rate 12 given in the problem, ft/h.

Copyright © Ed2Net Learning, Inc. 13 Now you try! 1)A cyclist travels 56 miles in 4 hours. What is the cyclist’s speed in feet per second? Round your answer to the nearest tenth, and show that your answer is reasonable. 1 mile = 5280 feet

Copyright © Ed2Net Learning, Inc. 14 In the proportion a = c, b d the products a · d and b · c are called cross products. You can solve a proportion for a missing value by using the Cross Products Property. Cross Products Property WORDSNUMBERSALGEBRA In a proportion, cross products are equal. 2 = · 6 = 3 · 4 If a = c b d and b ≠ 0 and d ≠ 0, then ad = bc.

Copyright © Ed2Net Learning, Inc. 15 Solving Proportions A) 5 = 3 9 w Use cross products. w = 27 5 Solve each proportion. 5w = 27 Divide both sides by 5. 5 = 3 9 w 5 (w) = 9 (3) 5w =

Copyright © Ed2Net Learning, Inc. 16 B) 8 = 1 x Use cross products. 86 = x 96 = x + 10 Subtract 10 from both sides. 8 = 1 x (12) = 1 (x + 10)

Copyright © Ed2Net Learning, Inc. 17 Now you try! 1) -5 = y 2 8 2) g + 3 = Solve each proportion.

Copyright © Ed2Net Learning, Inc. 18 A scale is a ratio between two sets of measurements, such as 1 in : 5 mi. A scale drawing or scale model uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing. A scale written without units, such as 32 : 1, means that 32 units of any measure correspond to 1 unit of that same measure.

Copyright © Ed2Net Learning, Inc. 19 Solution: x · 1 = 100 (0.8) Let x be the actual distance. The actual distance is 80 mi. A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance? 1 in : 100 mi x = 80 Scale Drawings and Scale Models Use cross products to solve. Write the scale as a fraction. map = 1 in actual 100 mi 0.8 in = 1 in x 100 mi

Copyright © Ed2Net Learning, Inc. 20 Solution: 127 = 100x Let x be the distance on the map. The actual distance is 80 mi. B) The actual distance between Bryan- College Station and Galveston is 127 mi. What is this distance on the map? 1 in : 100 mi 1.27 = x Use cross products to solve. Write the scale as a fraction. map = 1 in actual 100 mi x = 1 in mi 127 = 100x 100 Since x is multiplied by 100, divide both sides by 100 to undo the multiplication.

Copyright © Ed2Net Learning, Inc. 21 1) 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? Now you try!

Copyright © Ed2Net Learning, Inc. 22 Assessment 1) The ratio of the sale price of a jacket to the original price is 3 : 4. The original price is $64. What is the sale price? 2) Find the unit rate. A computer’s fan rotates 2000 times in 40 seconds. 3) Find the unit rate. Twelve cows produce 224,988 pounds of milk. 4) Lydia wrote 1 pages of her science report in one 2 hour. What was her writing rate in pages per minute? 4

Copyright © Ed2Net Learning, Inc. 23 5) 3 = 1 z 8 Solve each proportion. 6) x = ) b = ) f + 3 = ) -1 = 3 5 2d 10) 3 = s

Copyright © Ed2Net Learning, Inc. 24 Rates, Ratios, and Proportions A ratio is a comparison of two quantities by division. The ratio of a to b can be written a:b or a, b where b ≠ 0. Ratios that name the same comparison are said to be equivalent. A statement that two ratios are equivalent, such as 1 = 2, is called a proportion Read the proportion 1 = x “1 is to 15 as x is to 675.” Let’s review

Copyright © Ed2Net Learning, Inc. 25 Using Ratios review The ratio of faculty members to students at a college is 1:15. There are 675 students. How many faculty members are there? Faculty = 1 Students 15 1 = x = x 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.

Copyright © Ed2Net Learning, Inc. 26 A rate is a ratio of two quantities with different units, such as or 34mi, 2gal Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as 34mi, 2gal or 17 mi/gal. You can convert any rate to a unit rate. review

Copyright © Ed2Net Learning, Inc. 27 Garry ate 53.5 hot dogs in 12 minutes to win a contest. Find the unit rate. Round your answer to the nearest hundredth. Finding Unit Rates 4.46 ≈ x Write a proportion to find an equivalent ratio with a second quantity of 1. Divide on the left side to find x = x 12 1 The unit rate is approximately 4.46 hot dogs per minute. review

Copyright © Ed2Net Learning, Inc. 28 A rate such as 12in., 1 ft in which the two quantities are equal but use different units, is called a conversion factor. To convert a rate from one set of units to another, multiply by a conversion factor. Conversion factor review

Copyright © Ed2Net Learning, Inc. 29 Conversion factor A) As you go deeper underground, the earth’s temperature increases. In some places, it may increase by 25°C per kilometer. What is this rate in degrees per meter? To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. 25°C × 1km 1km 1000m The rate is 0.025°C per meter °C 1m review

Copyright © Ed2Net Learning, Inc. 30 B) The dwarf sea horse Hippocampus zosterae swims at a rate of feet per hour. What is this speed in inches per minute? To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity ft × 12 in 1h 1ft The speed is inches per hour in. 1h Step1: Convert the speed to inches per hour. Next page  review

Copyright © Ed2Net Learning, Inc. 31 To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity in × 1 h 1h 60 min The speed is inches per minute in 1 min Step2: Convert this speed to 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 600 = 50 ft/h. This is close to the rate 12 given in the problem, ft/h. review

Copyright © Ed2Net Learning, Inc. 32 review In the proportion a = c, b d the products a · d and b · c are called cross products. You can solve a proportion for a missing value by using the Cross Products Property. Cross Products Property WORDSNUMBERSALGEBRA In a proportion, cross products are equal. 2 = · 6 = 3 · 4 If a = c b d and b ≠ 0 and d ≠ 0, then ad = bc.

Copyright © Ed2Net Learning, Inc. 33 Solving Proportions A) 5 = 3 9 w Use cross products. w = 27 5 Solve each proportion. 5w = 27 Divide both sides by 5. 5 = 3 9 w 5 (w) = 9 (3) 5w = review

Copyright © Ed2Net Learning, Inc. 34 B) 8 = 1 x Use cross products. 86 = x 96 = x + 10 Subtract 10 from both sides. 8 = 1 x (12) = 1 (x + 10) review

Copyright © Ed2Net Learning, Inc. 35 A scale is a ratio between two sets of measurements, such as 1 in : 5 mi. A scale drawing or scale model uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing. A scale written without units, such as 32 : 1, means that 32 units of any measure correspond to 1 unit of that same measure. review

Copyright © Ed2Net Learning, Inc. 36 Solution: x · 1 = 100 (0.8) Let x be the actual distance. The actual distance is 80 mi. A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance? 1 in : 100 mi x = 80 Scale Drawings and Scale Models Use cross products to solve. Write the scale as a fraction. map = 1 in actual 100 mi 0.8 in = 1 in x 100 mi review

Copyright © Ed2Net Learning, Inc. 37 Solution: 127 = 100x Let x be the distance on the map. The actual distance is 80 mi. B) The actual distance between Bryan- College Station and Galveston is 127 mi. What is this distance on the map? 1 in : 100 mi 1.27 = x Use cross products to solve. Write the scale as a fraction. map = 1 in actual 100 mi x = 1 in mi 127 = 100x 100 Since x is multiplied by 100, divide both sides by 100 to undo the multiplication. review

Copyright © Ed2Net Learning, Inc. 38 You did a great job today!