Weighted Averages

Vocabulary Weighted Average – The sum of the product of the number of units and the value per unit divided by the sum of the number of units Mixture Problems – Problems where two or more parts are combined into a whole Uniform Motion Problems – Problems where something moves at a certain speed

Pets Jeri likes to feed her cat gourmet cat food that costs $1 Pets Jeri likes to feed her cat gourmet cat food that costs $1.75 per pound. However, food at that price is too expensive so she combines it with cheaper cat food that costs $0.50 per pound. How many pounds of cheaper food should Jeri buy to go with 5 pounds of gourmet food, if she wants the price to be $1.00 per pound? Example 9-1a

Let w = the number of pounds of cheaper cat food. Make a table. Units (lb) Price per Unit Price Gourmet cat food Mixed cat food 5 $1.75 $8.75 w $0.50 0.5w 5 + w $1.00 1.00(5 + w) Example 9-1b

Price of gourmet cat food price of cheaper cat food Write and solve an equation using the information in the table. Price of gourmet cat food plus price of cheaper cat food equals price of mixed cat food. 8.75 0.5w 1.00(5 + w) Original equation Distributive Property Subtract 0.5w from each side. Simplify. Example 9-1c

Subtract 5.0 from each side. Simplify. Divide each side by 0.5. Simplify. Answer: Jerry should buy 7.5 pounds of cheaper cat food to be mixed with the 4 pounds of gourmet cat food to equal out to $1.00 per pound of cat food. Example 9-1d

Answer: Cheryl should buy 4.6 ounces of beads. Cheryl has bought 3 ounces of sequins that cost $1.79 an ounce. The seed beads cost $0.99 an ounce. How many ounces of seed beads can she buy if she only wants the beads to be $1.29 an ounce for her craft project? Answer: Cheryl should buy 4.6 ounces of beads. Example 9-1e

Auto Maintenance To provide protection against freezing, a car’s radiator should contain a solution of 50% antifreeze. Darryl has 2 gallons of a 35% antifreeze solution. How many gallons of 100% antifreeze should Darryl add to his solution to produce a solution of 50% antifreeze? Example 9-2a

Amount of Solution (gallons) Let g = the number of gallons of 100% antifreeze to be added. Make a table. 35% Solution 100% Solution 50% Solution Amount of Solution (gallons) Price 2 0.35(2) g 1.0(g) 2 + g 0.50(2 + g) Example 9-2b

Write and solve an equation using the information in the table. Amount of antifreeze in 35% solution plus amount of antifreeze in 100% solution equals amount of antifreeze in 50% solution. 0.35(2) 1.0(g) 0.50(2 + g) Original equation Distributive Property Subtract 0.50g from each side. Simplify. Example 9-2c

Subtract 0.70 from each side. Simplify. Divide each side by 0.50. Simplify. Answer: Darryl should add 0.60 gallons of 100% antifreeze to produce a 50% solution. Example 9-2d

Answer: of a pound of 75% peanuts should be used. A recipe calls for mixed nuts with 50% peanuts. pound of 15% peanuts has already been used. How many pounds of 75% peanuts needs to be add to obtain the required 50% mix? Answer: of a pound of 75% peanuts should be used. Example 9-2e

To find the average speed for each leg of the trip, rewrite . Air Travel Mirasol took a non-stop flight from Newark to Austin to visit her grandmother. The 1500-mile trip took three hours and 45 minutes. Because of bad weather, the return trip took four hours and 45 minutes. What was her average speed for the round trip? To find the average speed for each leg of the trip, rewrite . Example 9-3a

Going Returning Example 9-3b

Definition of weighted average Round Trip Simplify. Answer: The average speed for the round trip was about 343.9 miles per hour. Example 9-3c

In the morning, when traffic is light, it takes 30 minutes to get to work. The trip is 15 miles through towns. In the afternoon when traffic is a little heavier, it takes 45 minutes. What is the average speed for the round trip? Answer: The average speed for the round trip was about 23 miles per hour. Example 9-3d

Rescue A railroad switching operator has discovered that two trains are heading toward each other on the same track. Currently, the trains are 53 miles apart. One train is traveling at 75 miles per hour and the other train is traveling at 40 miles per hour. The faster train will require 5 miles to stop safely, and the slower train will require 3 miles to stop safely. About how many minutes does the operator have to warn the train engineers to stop their trains? Example 9-4a

Draw a diagram. 53 miles apart Takes 5 miles to stop Example 9-4b

Let m = the number of minutes that the operator has to warn the train engineers to stop their trains safely. Make a table. Fast train Other train r d = rt t 75 m 75m 40 m 40m Example 9-4c

Distance traveled by fast train distance traveled by other train Write and solve an equation using the information in the table. Distance traveled by fast train plus distance traveled by other train equals 45 miles. 75m 40m 45 Original equation Simplify. Divide each side by 115. Example 9-4d

Round to the nearest hundredth. Convert to minutes by multiplying by 60. Answer: The operator has about 23 minutes to warn the engineers. Example 9-4e

Answer: They will be 7.5 miles apart in about 14 minutes. Two students left the school on their bicycles at the same time, one heading north and the other heading south. The student heading north travels 15 miles per hour, and the one heading south travels at 17 miles per hour. About how many minutes will they be 7.5 miles apart? Answer: They will be 7.5 miles apart in about 14 minutes. Example 9-4f

Homework Quiz

Coordinate Plane

Graphing Vocabulary Coordinate Plane – Plane containing the x and y axis. Also called the cartesian plane X-axis – Horizontal number line Y-axis – Vertical number line Origin – place where the axis intersect (0,0)

Graphing Vocabulary Quadrants – Starting in the upper left, moving counterclockwise I, II, III, IV X–coordinate – number corresponding to the value on the x-axis Y–coordinate – number corresponding to the value on the y-axis Ordered pair – coordinate values listed (x,y)

Write the ordered pair for point B. Follow along a horizontal line to find the x-coordinate on the x-axis. The x-coordinate is 3. Follow along a vertical line through the point to find the y-coordinate on the y-axis. The y-coordinate is –2. Answer: The ordered pair for point B is (3, –2). This can also be written as B(3, –2). Example 1-1a

Write the ordered pair for point C. Answer: (–4, 1) Example 1-1b

Use a table to help find the coordinates of each point. Write ordered pairs for points A, B, C, and D. Name the quadrant in which each point is located. Use a table to help find the coordinates of each point. Example 1-2a

Answer: A(–2, 2); II B(0, 2); none C(4, –2); IV D(–5, –4); III Point x-Coordinate y-Coordinate Ordered Pair Quadrant A –2 2 (–2, 2) II B (0, 2) None C 4 (4, –2) IV D –5 –4 (–5, –4) III Answer: A(–2, 2); II B(0, 2); none C(4, –2); IV D(–5, –4); III Example 1-2a

Answer: Q(–1, 3); II R(4, 1); I S(2, –4); IV T(–3, 0); none Write ordered pairs for points Q, R, S, and T. Name the quadrant in which each point is located. Answer: Q(–1, 3); II R(4, 1); I S(2, –4); IV T(–3, 0); none Example 1-2b

Plot A(3, 1) on the coordinate plane. Start at the origin. Move right 3 units since the x-coordinate is 3. Move up 1 unit since the y-coordinate is 1. Draw a dot and label it A. Example 1-3a

Plot B(–2, 0) on the coordinate plane. Start at the origin. Move left 2 units. Since the y-coordinate is 0, the point will be located on the x-axis. Draw a dot and label it B. Example 1-3b

Plot C(2, –5) on the coordinate plane. Start at the origin. Move right 2 units and down 5 units. Draw a dot and label it C. Example 1-3c

Plot each point on the coordinate plane. a. H(3, 5) b. J(0, 4) c. K(6, –2) Answer: Example 1-3d

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Homework – Write in Planner Section 3-9 Weighted Averages Section 4-1 The Coordinate Plane ALL