Ratio and Proportion

Objectives Use ratios and rates to solve real-life problems. Solve proportions.

Ratios A ratio is the comparison of two numbers written as a fraction. 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 games __ 7 = = games lost 3 games 3

Rates 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? ________ 120 miles ________ 20 miles Rate = = 60 gallons 1 gallon Your fuel efficiency is 20 miles per gallon.

Unit Analysis Writing the units when comparing each unit of a rate is called unit 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.

Proportion 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

Proportion 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 a:b::c:d = b d Extremes Means

Proportion 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

Proportion Example: Write the original proportion. Use the reciprocal property. Multiply both sides by 35 to isolate the variable, then simplify.

Proportion Example: Write the original proportion. Use the cross product property. Divide both sides by 6 to isolate the variable, then simplify.

You Try It! If the average person lives for 75 years, how long would that be in seconds?

You Try It! If the average person lives for 75 years, how long would that be in seconds? To solve this problem we need to convert 75 years to seconds. We can do this by breaking the problem down into smaller parts by converting years to days, days to hours, hours to minutes and minutes to seconds. There are 365.25 days in one year, 24 hours in one day, 60 minutes in 1 hour, and 60 seconds in a minute. Multiply the fractions, and use unit analysis to determine the correct units for the answer.

You Try It! John constructs a scale model of a building. He says that 3/4th feet of height on the real building is 1/5th inches of height on the model. What is the ratio between the height of the model and the height of the building? If the model is 5 inches tall, how tall is the actual building in feet?

You Try It! What is the ratio between the height of the model and the height of the building? What two pieces of information does the problem give you to write a ratio? For every 3/4th feet of height on the building… the model has 1/5th inches of height. Therefore the ratio of the height of the model to the height of the building is… This is called a scale factor.

You Try It! If the model is 5 inches tall, how tall is the actual building in feet? To find the actual height of the building, use the ratio from the previous step to write a proportion to represent the question above. Use the cross product. Isolate the variable, then simplify. Don’t forget your units.