Grade 8 Pre-Algebra Rates, Ratios, and Proportions CONFIDENTIAL

Factor each trinomial by guess and check: Warm Up Factor each trinomial by guess and check: 1) 2c - 5 = c + 4 1) c = 9 2) r = 1 2) 8r + 4 = 10 + 2r 3) x = 12 3) 2x -1 = x + 11 4) 28 - 0.3y = 0.7y - 12 4) y = 40 CONFIDENTIAL

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. 12 24 Read the proportion 1 = x 15 675 “1 is to 15 as x is to 675.” CONFIDENTIAL

There are 45 faculty members. 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 Write a ratio comparing faculty to students. 1 = x 15 675 Write a proportion. Let x be the number of faculty members. 1 = x 15 675 675 Since x is divided by 675, multiply both sides of the equation by 675. x = 45 There are 45 faculty members. CONFIDENTIAL

1) The ratio of games won to games lost for a baseball team is 3 : 2. 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? 1) 12 CONFIDENTIAL

or 17 mi/gal. You can convert any rate to a unit rate. 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. CONFIDENTIAL

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

1) Cory earns $52.50 in 7 hours. Find the unit rate. Now you try! 1) Cory earns $52.50 in 7 hours. Find the unit rate. 1) 7.5 CONFIDENTIAL

To convert a rate from one set of units Conversion factor 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. CONFIDENTIAL

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? 25°C × 1km 1km 1000m To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. 0.025°C 1m The rate is 0.025°C per meter. CONFIDENTIAL

Step1: Convert the speed to inches per hour. B) The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. What is this speed in inches per minute? Step1: Convert the speed to inches per hour. To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity. 52.68ft × 12 in 1h 1ft 632.16 in. 1h The speed is 632.16 inches per hour. Next page  CONFIDENTIAL

Step2: Convert this speed to inches per minute. 632.16 in × 1 h 1h 60 min To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. 10.536 in 1 min The speed is 10.536 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, 52.68 ft/h. CONFIDENTIAL

Now you try! 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 1) 20.53 ft/sec CONFIDENTIAL

the products a · d and b · c are called cross products. Cross Products Property 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. WORDS NUMBERS ALGEBRA In a proportion, cross products are equal. 53.5 = x 12 1 2 · 6 = 3 · 4 If 53.5 = x and b ≠ 0 and d ≠ 0, then ad = bc. CONFIDENTIAL

Solving Proportions Solve each proportion. A) 5 = 3 9 w 5 = 3 9 w 5 = 3 9 w 5 (w) = 9 (3) Use cross products. 5w = 27 5w = 27 5 5 Divide both sides by 5. w = 27 5 CONFIDENTIAL

B) 8 = 1 x + 10 12 8 = 1 x + 10 12 8 (12) = 1 (x + 10) Use cross products. 96 = x + 10 -10 -10 Subtract 10 from both sides. 86 = x CONFIDENTIAL

Now you try! Solve each proportion. 1) -5 = y 2 8 1) y = -20 2 8 1) y = -20 2) g + 3 = 7 5 4 2) g = 5.75 CONFIDENTIAL

A map is an example of a scale drawing. 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. CONFIDENTIAL

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

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 Solution: map = 1 in actual 100 mi Write the scale as a fraction. x = 1 in 127 100 mi Let x be the distance on the map. 127 = 100x Use cross products to solve. 127 = 100x 100 100 Since x is multiplied by 100, divide both sides by 100 to undo the multiplication. 1.27 = x The actual distance is 80 mi. CONFIDENTIAL

Now you try! 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? 1) 0.5 inch CONFIDENTIAL

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? 1) $48 2) 50 times/sec 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. 3) 18749 pounds of milk/cow 4) Lydia wrote 1 pages of her science report in one 2 hour. What was her writing rate in pages per minute? 4 4) 3 pages per minute 40 CONFIDENTIAL

Solve each proportion. 5) 3 = 1 z 8 8) f + 3 = 7 12 2 5) z = 24 5) 3 = 1 z 8 8) f + 3 = 7 12 2 5) z = 24 8) f = 39 6) x = 1 3 5 9) -1 = 3 5 2d 6) x = 0.6 9) d = -7.5 7) b = 3 4 2 10) 3 = s - 2 14 21 7) b = 6 10) s = 6.5 CONFIDENTIAL

Rates, Ratios, and Proportions Let’s review 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. 12 24 Read the proportion 1 = x 15 675 “1 is to 15 as x is to 675.” CONFIDENTIAL

There are 45 faculty members. review 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 Write a ratio comparing faculty to students. 1 = x 15 675 Write a proportion. Let x be the number of faculty members. 1 = x 15 675 675 Since x is divided by 675, multiply both sides of the equation by 675. x = 45 There are 45 faculty members. CONFIDENTIAL

or 17 mi/gal. You can convert any rate to a unit rate. review 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. CONFIDENTIAL

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

To convert a rate from one set of units review Conversion factor 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. CONFIDENTIAL

review 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? 25°C × 1km 1km 1000m To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. 0.025°C 1m The rate is 0.025°C per meter. CONFIDENTIAL

Step1: Convert the speed to inches per hour. review B) The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. What is this speed in inches per minute? Step1: Convert the speed to inches per hour. To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity. 52.68ft × 12 in 1h 1ft 632.16 in. 1h The speed is 632.16 inches per hour. Next page  CONFIDENTIAL

Step2: Convert this speed to inches per minute. review Step2: Convert this speed to inches per minute. 632.16 in × 1 h 1h 60 min To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. 10.536 in 1 min The speed is 10.536 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, 52.68 ft/h. CONFIDENTIAL

the products a · d and b · c are called cross products. review Cross Products Property 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. WORDS NUMBERS ALGEBRA In a proportion, cross products are equal. 53.5 = x 12 1 2 · 6 = 3 · 4 If 53.5 = x and b ≠ 0 and d ≠ 0, then ad = bc. CONFIDENTIAL

review Solving Proportions Solve each proportion. A) 5 = 3 9 w 5 = 3 5 = 3 9 w 5 (w) = 9 (3) Use cross products. 5w = 27 5w = 27 5 5 Divide both sides by 5. w = 27 5 CONFIDENTIAL

review B) 8 = 1 x + 10 12 8 = 1 x + 10 12 8 (12) = 1 (x + 10) 8 = 1 x + 10 12 8 (12) = 1 (x + 10) Use cross products. 96 = x + 10 -10 -10 Subtract 10 from both sides. 86 = x CONFIDENTIAL

A map is an example of a scale drawing. review 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. CONFIDENTIAL

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

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

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