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Section 1.3 and intro to 1.4 (Proportions) 1. Use unit rates and dimensional analysis to solve real-life problems. 2. Begin to solve proportions. A ratio is the ________________________________. For example: Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses? Because we are comparing wins to losses the first number in our ratio should be the number of wins and the second number is the number of losses. The ratio is ___________ games won ______ 7 = 7 games = __ games lost 3 games 3 In a ratio, if the numerator and denominator are measured in different units then the ratio is called a rate. A unit rate is a rate per one given unit, like 60 miles per 1 hour. Example: You can travel 120 miles on 60 gallons of gas. What is your fuel efficiency in miles per gallon? ________ miles = ________ Rate = 120 60 gallons 1 gallon Your fuel efficiency is _______ miles per gallon. Notes: Convert Rates Customary Units of Measure Smaller Larger 12 inches 1 foot 16 ounces 1 pound 8 pints 1 gallon 3 feet 1 yard 5,280 feet 1 mile Notes: Convert Rates Metric Units of Measure Smaller Larger 100 centimeters 1 meter 1,000 grams 1 kilogram 1,000 milliliters 1 liter 10 milliliters 1 centimeter 1,000 milligrams 1 gram Notes: Convert Rates Each of the relationships in the tables can be written as a _____________. Like a unit rate, a unit ____ is one in which the denominator is 1 unit. Below are three examples of unit ratios: 12 inches 1 foot 16 ounces 100 centimeters 1 pound 1 meter Notes: Convert Rates The ______ and __________ of each of the unit ratios shown are equal. So, the value of each ratio is ______. You can convert one rate to an equivalent rate by multiplying by a unit ratio or its reciprocal. When you convert rates, you include the _______ in your computation. The process of including units of measure as factors when you compute is called ____________. Writing the units when comparing each unit of a rate is called dimensional analysis. You can multiply and divide units just like you would multiply and divide numbers. When solving problems involving rates, you can use unit analysis to determine the correct units for the answer. Example: How many minutes are in 5 hours? 5 hours • 60 ________ minutes = 300 minutes 1 hour To solve this problem we need a unit rate that relates minutes to hours. Because there are 60 minutes in an hour, the unit rate we choose is 60 minutes per hour. Notes: Convert Rates Example: A remote control car travels at a rate of 10 feet per second. How many inches per second is this? Steps: 10 ft = 10 ft 12 in Use 1 foot=12 inches 1s 1s 1 ft = 10 ft 1s 12 in 1 ft Divide out common units Notes: Convert Rates = 10 1s 12 in 1 = 120 in 1s Simplify Simplify So, 10 feet per second equals 120 inches per second. Dimensional Analysis Examples 1. A gull can fly at a speed of 22 miles per hour. About how many feet per hour can a gull fly? (Use the chart) Essential Question Explain why the ratio 3 feet has a value of 1. 1 yard ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ Dimensional Analysis Examples 2. An AMTRAK train travels 125 miles per hour. Convert the speed to miles per minute. Round to the nearest tenth. (Use the chart) An equation in which two ratios are equal is called a proportion. A proportion can be written using colon notation like this a:b::c:d or as the more recognizable (and useable) equivalence of two fractions. ___ a = ___ c b d When Ratios are written in this order, a and d are the extremes, or outside values, of the proportion, and b and c are the means, or middle values, of the proportion. ___ a = ___ c b d a:b::c:d Extremes Means To solve problems which require the use of a proportion we can use one of two properties. The reciprocal property of proportions. If two ratios are equal, then their reciprocals are equal. The cross product property of proportions. The product of the extremes equals the product of the means Example: 5 35 3 x Write the original proportion. 3 x 5 35 Use the reciprocal property. 3 x Multiply both sides by 35 to isolate 35 35 the variable, then simplify. 5 35 21 x Example: 2 6 x 9 Write the original proportion. __________ _ Use the cross product property. _________ ______ x Divide both sides by 6 to isolate the variable, then simplify.