4.1 Ratios Ratio: uses division to compare two numbers. Numbers are usually the same unit of measurement. Do not convert to a decimal or a mixed fraction, denominator not equal to 0. Read as “the ratio of a to b.” Written as “a to b”, a:b or a/b 6M 6R 6W Boys 9 9 8 Girls 10 10 11 What is the ratio of girls to boys in 6M? What is the ratio of 6R to 6W in our math class? In our math class, what is the ratio of 6M students to all students?

4.2 Rates Rate: is a ratio that compare two quantities measured in different units, which is usually expressed as a Unit Rate. Unit Rate: is a rate whose denominator is 1. Write the rate as a fraction and then divide the numerator and denominator by the denominator. Show the units of measure. Word after “per” is in the denominator. If it drove 150 miles to Washington, DC in 2 hours, what was my speed? (in miles per hour) If I drove 90 miles to New York in 1 hour and 40 minutes, what was my speed? (in miles per hour)

4.3 Identifying and Writing Proportions Equivalent ratio: name the same comparison Proportion: an equation that states that two ratios are equivalent. Same Denominator We can use equivalent ratios (fractions) to solve proportions where one number is unknown. Are 3/5 and 12/20 equivalent ratios? Common Denominator is 20. 3/5 × 4/4 = 12/20 They are equivalent. Are 1/3 and 17/48 equivalent ratios? Common denominator is 48. 1/3 × 16/16 = 16/48. They are not equivalent.

4.4 Solving Proportions There are at least three ways to solve a proportion: equivalent fractions and cross products, algebra What is an equivalent fraction to ¾ = x/132 ? Equivalent Fractions ¾ = x/132 What times 4 is 132? (33) ¾ · 33/33 = 99/132 Cross Products 3 · 132 = 4 · x → 396 = 4x → 396 ÷ 4 = 99 Algebra: What operation is happening? ¾ · 132 = ¾ · 132/1 = 396/4 = 99

4.7 Dimensional Analysis Unit conversion factor is a fraction where the numerator and denominator represent the same quantity in different units. We can use these fractions as proportions to convert measurement. Customary Metric Measurement Measurement 5,280 feet/1 mile 1,000 meters/1 kilometer 12 inches/1 foot 100 cm/1 meter 16 ounces/1 pound 1,000 grams/1 kilogram 2000 pounds/1 ton 1,000 milligrams/1 gram 8 fluid ounces/1 cup 1,000 liters/1 kiloliter 2 pints/1 quart 100 centiliters/1 liter

4.7 Dimensional Analysis Other measurements 12 months/1 ______ 7 ____/1 _______ 60_______/1 hour 3,600 _____/1 ______ 52 ______/_______ 24 _____/1 _____ 4 ____/1 gallon 12 ______/ 1 foot 1,760 _____/1 mile Approximate conversions 1.609 km/1 mi 0.62 mi/1 km 1.0567 L/1 qt 0.95 qt/1 L 2.54 cm/1 inch 0.39 in/1 cm 0.914 m/1 yd 1.09 yd/1 m .454 kg/1 pound 2.2 lb/1 kg

4.8 Similar Figures and Proportions Similar figures have the same shape, but not necessarily the same size. The symbol “~” means “is similar to.” Similar figures may be reflections or flips of each other Corresponding angles and sides of two or more similar polygons are in the same relative position. Similar figures must have: Equal corresponding angles Corresponding sides with the same proportional lengths You can use proportions to figure out the dimensions of similar figures.

4.9 Using Similar Figures Indirect measurement is a method for using proportions to find an unknown length or distance in similar figures. Examples: surveyors, golf range finders, GPS

4.9 Using Similar Figures Location Length of A Length of B Slide 12 cm

4.10 Scale Drawings and Scale Models Scale Drawing: proportional two-dimensional drawing of an object. Scale Model: proportional three-dimensional model of an object Scale Factor: the dimensions of the drawing or model to the actual, expressed as Model/Actual Examples: blueprints, maps, toys, sculptures EARTH Actual Globe Scale Diameter 7,920 miles 12 inches 1/41,817,600 Actual Map Scale Circumference 24,901 miles 48 inches 1/32,869,320

4.10 Scale Drawings and Scale Models Diameter Model (Globe): 1 foot Actual 7,926 miles × 5,280 feet/mile = 41,817,600 Scale Model/Actual: 1/41,817,600 Circumference Model (World Map): 48 inches Actual: 24,901 mi × 5,280 feet/mile × 12 inches/foot = 1,577,727,360 inches Model/Actual: 48/1,577,727,360 = 1/32,869,320

4.10 Scale Drawings and Scale Models Clothespin actual: 7.3 cm Clothespin sculpture (model): 14 m Scale: