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UNIT 3: RATIOS AND PROPORTIONAL RELATIONSHIPS MCC7.RP.1 MCC7.RP.2a MCC7.RP.2b MCC7.RP.2c MCC7.RP.2d MCC7.RP.3 MCC7.G.1 LESSON 1: RATIOS Please fill in your guided notes as you view the presentation. Have fun!! IN THE REAL WORLD Baseball How can you compare a baseball team’s wins to its losses during spring training? A ratio uses division to compare two numbers. There are three ways to write a ratio of two numbers. WRITING A RATIO All three ways of writing the ratio of two numbers are read “the ratio of a to b,” so 18 : 13 is read “the ratio of eighteen to thirteen.” Two ratios are equivalent ratios when they have the same value. EXAMPLE 1 WRITING A RATIO 1. You can make comparisons about games played by the Cubs. a. Wins to losses wins = 17, losses = 14 Answer: _______ b. Wins to games played wins = 17, games = 17 + 14 = 31 Answer: _________ YOUR TURN Use the table above to write the ratio for the Padres. 1. Wins to losses 2. Wins to games played 3. Losses to wins EXAMPLE 2 WRITING RATIOS IN SIMPLEST FORM Amusement Parks A ride on a roller coaster lasts 2 minutes. Suppose you wait in line for 1 ½ hours to ride the roller coaster. Follow the steps below to find the ratio of time spent in line to time spent on the ride. -Write hours as minutes so that the units are the same. -Write the ratio of time spent in line to time spent on the ride. EXAMPLE 3 COMPARING RATIOS 1. 2. Music Luis and Amber compared their CD collections. To determine who has a greater ratio of rock CDs to pop CDs, write the ratios. Write ratios as fractions (rock/pop). Write fractions as decimals and compare. Luis: Amber: YOUR TURN 1. Does Luis or Amber have a greater ratio of pop CDs to hip-hop CDs? 1. Does Luis or Amber have a greater ratio of hiphop CDs to rock CDs? LESSON 2: RATES A rate is a ratio of two quantities measured in different units. A unit rate is a rate that has a denominator of 1 unit. The three unit rates below are equivalent. In the third rate, “per” means “for every.” EXAMPLE 1 FINDING A UNIT RATE Kudzu During peak growing season, the kudzu vine can grow 6 inches in 12 hours. What is the growth rate of kudzu in inches per hour? Solution First, write a rate comparing the inches grown to the hours it took to grow. Then rewrite the fraction so that the denominator is 1. ANSWER The growth rate of kudzu is about 0.5 inch per hour. YOUR TURN: FIND THE UNIT RATE 1. 2. 3. 1. $54 in 6 hours 68 miles in 4 days 2 cups in 8 servings Average Speed If you know the distance traveled and the travel time for a moving object, you can find the average rate, or average speed, by dividing the distance by the time. EXAMPLE 2 FINDING AN AVERAGE SPEED Speed Skating A skater took 2 minutes 30 seconds to complete a 1500 meter race. What was the skater’s average speed? Rewrite the time so that the units are the same. 2 min + 30 sec = 120 sec + 30 sec = 150 sec Find the average speed. EXAMPLE 3 COMPARING UNIT RATES Pasta A store sells the same pasta the following two ways: 10 pounds of bulk pasta for $15.00 and 2 pounds of packaged pasta for $3.98. To determine which is the better buy, find the unit price for both types. ANSWER The bulk pasta is the better buy because it costs less per pound. YOUR TURN 1. It takes you 1 minute 40 seconds to walk 550 feet. What is your average speed? 2. Which of the following is the better buy: 2 AA batteries for $1.50 or 6 AA batteries for $4.80? LESSON 3: SLOPE & UNIT RATES The slope of a nonvertical line is the ratio of the rise (vertical change) to the run (horizontal change) between any two points on the line, as shown below. A line has a constant slope. EXAMPLES OF LINES WITH POSITIVE, NEGATIVE, AND ZERO SLOPES ARE SHOWN The slope of a vertical line is undefined EXAMPLE 1 FINDING THE SLOPE OF A LINE a. b. To find the slope of a line, find the ratio of the rise to the run between two points on the line EXAMPLE 2 INTERPRETING SLOPE AS A UNIT RATE Slope as a rate – When the graph of a line represents a real-world situation, the slope of the line can often be interpreted as a rate. Volcanoes The graph represents the distance traveled by a lava flow over time. To find the speed of the lava flow, find the slope of the line. YOUR TURN 1. Plot the points (3, 4) and (6, 3). Then find the slope of the line that passes through the points. 2. In Example 2, suppose the line starts at the origin and passes through the point (3, 6). Find the speed of the lava flow. EXAMPLE 3 USING SLOPE TO DRAW A LINE Draw the line that has a slope of –3 and passes through (2, 5). 1. Plot (2,5) 2. Write the slope as a fraction 3. Move 1 unit to the right, and 3 units down to plot the second point 4. Draw a line through the two points YOUR TURN 3. Draw the line that has a slope of 1/3 and passes through (2, 5). LESSON 4: WRITING AND SOLVING PROPORTIONS Sports A person burned about 150 calories while skateboarding for 30 minutes. About how many calories would the person burn while skateboarding for 60 minutes? In Example 1, you will use a proportion to answer this question. A proportion is an equation that states that two ratios are equivalent. Algebra = , where b and d are nonzero numbers. EXAMPLE 1 USING EQUIVALENT RATIOS Sports A person burned about 150 calories while skateboarding for 30 minutes. About how many calories would the person burn while skateboarding for 60 minutes? In Example 1, you will use a proportion to answer this question. To find the number C of calories the person would burn while skateboarding for 60 minutes, solve the proportion = . Answer: Ask yourself, what number can you multiply by 30 to get 60? The answer is 2. So, multiply 150 x 2 to get 300. So the person would burn about 300 calories while skateboarding for 60 minutes. EXAMPLE 2 SOLVING PROPORTIONS USING ALGEBRA Solve the proportion = . . YOUR TURN 1. Use equivalent ratios to solve the proportion. 5. 6. 2. 7. 3. 8. 4. Use algebra to solve the proportion. EXAMPLE 3 WRITING AND SOLVING A PROPORTION Setting Up a Proportion There are different ways to set up a proportion. Consider the following problem. Yesterday you bought 8 bagels for $4. Today you want only 5 bagels. How much will 5 bagels cost? EXAMPLE 3 WRITING AND SOLVING A PROPORTION Empire State Building Step 1: Write a proportion Step 2: Solve the proportion At maximum speed, the elevators in the Empire State Building can pass 80 floors in 45 seconds. Follow the steps below to find the number of floors that the elevators can pass in 9 seconds. LESSON 5: SOLVING PROPORTIONS USING CROSS PRODUCTS In the Real World Science At space camp, you can sit in a chair that simulates the force of gravity on the moon. A person who weighs 105 pounds on Earth would weigh 17.5 pounds on the moon. How much would a 60 pound dog weigh on the moon? You’ll find the answer in Example 2. In the proportion , the products 2 • 6 and 3 • 4 are called cross products. Notice that the cross products are equal. This suggests the following property. CROSS PRODUCTS PROPERTY Cross Products Property Words The cross products of a proportion are equal. EXAMPLE 1 SOLVING A PROPORTION USING CROSS PRODUCTS Use the cross products property to solve. EXAMPLE 2 WRITING AND SOLVING A PROPORTION Science At space camp, you can sit in a chair that simulates the force of gravity on the moon. A person who weighs 105 pounds on Earth would weigh 17.5 pounds on the moon. How much would a 60 pound dog weigh on the moon? EXAMPLE 3 WRITING AND SOLVING A PROPORTION Penguins At an aquarium, the ratio of rockhopper penguins to African penguins is 3 to 7. If there are 50 penguins, how many are rockhoppers? First, determine the ratio of rockhoppers to total penguins. Then, set up proportion. YOUR TURN In John’s class, the ratio of boys to girls is 5 to 8. If there are 39 students in his class, how many are girls? LESSON 6: SCALE DRAWINGS AND MODELS The floor plan below is an example of a scale drawing. A scale drawing is a diagram of an object in which the dimensions are in proportion to the actual dimensions of the object. The scale on a scale drawing tells how the drawing’s dimensions and the actual dimensions are related. The scale “1 in. : 12 ft” means that 1 inch in the floor plan represents an actual distance of 12 feet. EXAMPLE 1 USING THE SCALE OF A MAP Maps Use the map of Maine to estimate the distance between the towns of China and New Sweden. Solution From the map’s scale, 1 centimeter represents 65 kilometers. On the map, the distance between China and New Sweden is 4.5 centimeters. Find the distance in kilometers. EXAMPLE 2 FINDING A DIMENSION ON A SCALE MODEL White House A scale model of the White House appears in Tobu World Square in Japan. The scale used is 1 : 25. The height of the main building of the White House is 85 feet. Find this height on the model. Write and solve a proportion to find the height h of the main building of the model of the White House. EXAMPLE 3 FINDING THE SCALE Dinosaurs A museum is creating a full-size Tyrannosaurus rex from a model. The model is 40 inches in length, from the nose to the tail. The resulting dinosaur will be 40 feet in length. What is the model’s scale? Write a ratio. Make sure that both measures are in inches. Then simplify the fraction. ANSWER The model’s scale is 1 : 12. YOUR TURN 1. The model of the Eiffel Tower in Tobu World Square is 12 meters high. The scale used is 1 : 25. Estimate the actual height of the Eiffel Tower. 2. On a map of Colorado, the distance from Rico to Lizard Head Pass on Route 145 is about 9.5 cm. From the map scale, 1 cm represents 2 km. Estimate the actual distance between the towns. 3. The caboose on a model train is 6.75 inches long. The full-size caboose is 36 feet. What is the model’s scale?