1. Use ratios and rates to solve real-life problems.2. Solve proportions.
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
In a ratio, if the numerator and denominator are measured indifferent 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? Rate = 120 miles = ________ ________ 20 miles 60 gallons 1 gallon Your fuel efficiency is 20 miles per gallon.
Writing the units when comparing each unit of a rate is called unitanalysis.You can multiply and divide units just like you would multiply anddivide numbers. When solving problems involving rates, you canuse 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 hourTo solve this problem we need a unit rate that relates minutes tohours. Because there are 60 minutes in an hour, the unit rate wechoose is 60 minutes per hour.
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:dor 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, oroutside values, of the proportion, and b and c are the means, ormiddle values, of the proportion. ___ = ___ a c a:b::c:d b d Extremes Means
To solve problems which require the use of a proportion we can use oneof 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 = Write the original proportion. 3 x 3 x = Use the reciprocal property. 5 35 3 x Multiply both sides by 35 to isolate 35 • = • 35 5 35 the variable, then simplify. 21 = x
Example: 2 6 = Write the original proportion. x 9 9•2 = 6• x Use the cross product property. 18 6 x Divide both sides by 6 to isolate the = 6 6 variable, then simplify. 3= x
If the average person lives for 75 years, how long would that bein seconds?
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. 365.25 days 24 hours 60 minutes 60 seconds75 years • • • • = 2366820000 1 year 1 day 1 hour 1 minute seconds Multiply the fractions, and use unit analysis to determine the correct units for the answer.
John constructs a scale model of a building. He says that 3/4thfeet of height on the real building is 1/5th inches of height onthe model.What is the ratio between the height of the model and theheight of the building?If the model is 5 inches tall, how tall is the actual building infeet?
What is the ratio between the height of the model and theheight of the building?What two pieces of information does the problem give youto 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 heightof the building is… 1 inches 1 4 4 inches 5 = • = This is called a scale factor. 3 feet 5 3 15 feet 4
If the model is 5 inches tall, how tall is the actual building infeet? To find the actual height of the building, use the ratio from the previous step to write a proportion to represent the question above. 4 inches 5 inches = 15 feet x 4 • x = 5 • 15 Use the cross product. 4 x 75 = Isolate the variable, then simplify. 4 4 x = 18.75 feet Don’t forget your units.