LESSON 8.1: RATIOS AND PROPORTIONS OBJECTIVE: To write ratios and solve proportions Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Definition: Ratio an expression that A ratio is ____________________ ____________________________. If a and b are two numbers, the ratio of a to b can be written in the following ways: compares two quantities by division a/b a:b a is to b Mrs. McConaughy Geometry

Mrs. McConaughy Geometry FINDING RATIOS: Determine the following ratios: female to male students in this classroom ____ male to female students male to total students total students to female students in this classroom ____ Mrs. McConaughy Geometry

Mrs. McConaughy Geometry FINDING RATIOS: Determine the following ratios: Ratio of ♥ to ♣:__ ♥♣♥♣♥♣♥♣♥♣♥♣♥♣♣ ♣♣♣♥♥♣♣♣♥♥♣♣♣♥♥ ♣♥♣♣♣♥♣♣♣♥♣♣♥♥♥ ♥♥♥♣♣♥♥♥♥♥♥♣♣♥♥♥ Ratio of shaded to unshaded area:__ Ratio of shaded area to area of the whole figure:__ Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Example A scale model of a car is 4 inches long. The actual car is 15 ft. long. a.) What is the ratio of the length of the model to the length of the car? ________________ b.) What is the ratio of the length of the car to the length of the model? ______________ Write both measurements in the same units. NOTE: ORDER is important when writing ratios. Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Example A photo that is 8 inches wide and 5 1/3 inches high is enlarged to poster size which is 2 ft. wide by 1 1/3 ft. high. What is the ratio of the height of the photo to the height of the poster? . NOTE: ORDER is important when writing ratios. Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Proportion When two ratios are equal, you have a ________. proportion Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Proportion a statement of equality between two ratios (equal ratios). A proportion is___________ _______________________ _______________________. extremes a : b = c : d means Mrs. McConaughy Geometry

Mrs. McConaughy Geometry PROPERTIES OF PROPORTIONS 1. Cross Product Property The product of the extremes equals the product of the means. If a = c, then________. b d 2. Reciprocal Property If two ratios are equal, then their reciprocals are also equal. If a = c, then b d __________. ad = bc b = d a c Mrs. McConaughy Geometry

To solve the proportion, you find the value of the variable. Solve the proportions. a. 4 = 5 b. 3 = 2 x 7 y +2 y Mrs. McConaughy Geometry

Example Solving for a Variable Before you cross multiply, determine if you can reduce fractions. Example: 26 = __ 50 13 25 How can you determine the answer without cross multiplying? Mrs. McConaughy Geometry

Additional Properties of Proportions Two more properties of proportions, which are especially useful in geometry, are: b d c + d d Mrs. McConaughy Geometry

Mrs. McConaughy Geometry NOTE: You can avoid memorizing many of the Properties of Proportions by remembering to write your ratio (comparison of two quantities by division) in words before setting up and solving your proportion by cross multiplying. ORDER is important when setting up proportions. Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Complete: The Cross-Product Property states that________________. Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Solving for a Variable 3x = 2 (x + 1) 3x = 2x + 2 x = 2 Mrs. McConaughy Geometry

Mrs. McConaughy Geometry USING PROPORTIONS If you earn $380 working for two weeks, how much will you earn in 15 weeks? KEY: Words  Symbols Always remember to set up your proportion in words first! Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Example: (Always ask: What two quantities am I comparing?) 380 x 2 15 2 15 380 x Mrs. McConaughy Geometry

Mrs. McConaughy Geometry 2x = (380)(15) 2x = 5700 x = 2850 You will earn $2,850 in 15 weeks. Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Extended Proportions An extended proportion is _______________________. Mrs. McConaughy Geometry

Mrs. McConaughy Geometry USING PROPORTIONS The floor plan of a house is drawn to the scale of ¼ inch = 1 foot. The master bedroom measures 3 inches by 3 ¾ inches on the blueprints. What are the actual measurements of the room? Mrs. McConaughy Geometry

The actual size of the room is_______________ . 12 feet by 15 feet Pick your poison: Choose how you want to set up your two ratios in words first! NOTE: Rather than writing an extended proportion, you may find it easier to write two separate proportions. Mrs. McConaughy Geometry

Final Checks for Understanding 1. A ratio is_______________________. 2. A proportion is ___________________. 3. What is the most efficient way to solve 8 = x ? Explain. 24 9 4. Solve for x: 4 = x x 16. Mrs. McConaughy Geometry

Summary: Properties of Proportions d c ac b d c + d d Mrs. McConaughy Geometry

Mrs. McConaughy Geometry Homework Assignment: Ratios and Proportions WS Mrs. McConaughy Geometry