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# Ratios And Proportions Notes

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### Ratios And Proportions Notes

1. 1. Sapere Aude “ Dare to Know”
2. 2. Spot the Error <ul><ul><ul><ul><ul><li>x = y </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>x 2 = (x*y) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>x 2 - y 2 = (x*y) - y 2 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>(x+y) (x-y) = y(x-y) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>(x+y) = y </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>x+x = x </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>2x = x </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>2 = 1 </li></ul></ul></ul></ul></ul>
3. 3. Math Magic! <ul><li>Start with any number. </li></ul><ul><li>Add 12. </li></ul><ul><li>Multiply by 4. </li></ul><ul><li>Subtract 8. </li></ul><ul><li>Divide by 2. </li></ul><ul><li>Subtract your original number. </li></ul><ul><li>Subtract 6. </li></ul><ul><li>Subtract your original number. </li></ul><ul><li>Divide by 2. </li></ul>And the surprise answer is…
4. 4. Why Learn Math? <ul><li>\$\$\$ (Careers that make ‘bank’) </li></ul><ul><li>Pure enjoyment </li></ul><ul><li>Applicability in everyday life </li></ul><ul><li>Develop ability to think LOGICALLY! </li></ul>Jordan Woy, a highly respected sports agent and a principal in the sports marketing/management firm of Schlegel Sports . Jordan has represented numerous high profile athletes Here is what Jordon had to say: I think there are several reasons why so many athletes “go broke”. First, whether it is a lottery winner, an athlete or a star entertainer, if they are not equipped with the knowledge on how to make and save money they are in trouble. When they didn’t earn it through disciplined business practices and they don’t have those skills they usually go through it quickly. Most lottery winners or athletes make a great deal of money in a short period of time. They start spending it on things that only go down in value (cars, jewelry, partying, entourage, etc) and start to evaporate the money they do have. They can carry this off until they stop earning big money. This is when the trouble starts. It is hard to believe that MC Hammer, Mike Tyson, Evander Holyfield and now Ed McMahon are broke… Most athletes play for four to ten years if they are lucky. After they pay taxes (can be 40 to 50%) and agent fees and buy their first homes, cars, outfits, jewelry (plus, cars, clothes and jewelry for friends and family), they are left with very little…. However, if athletes educate themselves, learn money management skills and make smart, safe investments along the way, they are usually in very good shape.
5. 5. So what will we be learning in this class? <ul><li>The difference between ratios and proportions and their application </li></ul><ul><li>A little geometry (you’ll be ahead of the high school game) </li></ul><ul><li>Basic probability theory…FUN stuff! (you seriously will love it, or your money back guaranteed) </li></ul>
6. 6. Ratio and Proportion
7. 7. Objectives <ul><li>Use ratios and rates to solve real-life problems. </li></ul><ul><li>Solve proportions. </li></ul>
8. 8. Ratios A ratio is the comparison of two numbers using division. 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 ___________ games lost = 7 games _______ 3 games = 7 __ 3
9. 10. Ratio Comparing two things by division A fraction is a type of ratio that compares the part to the whole 5 to 8 5:8 5/8
10. 11. Stats from Last Night’s Game <ul><li>Girls </li></ul><ul><li>Tigrett 35, Rose Hill 32 </li></ul><ul><li>Scoring: </li></ul><ul><li>T - Quinn Thomas 23, Sydni Spraggins 10, Beard 2; </li></ul><ul><li>RH - Jada Perkins 10, Magee 7, Beard 5, Fuller 4, Holt 4, Bates 2. </li></ul><ul><li>3-pointers: T - Spraggins; RH - Beard. </li></ul><ul><li>Halftime score: T 19-17. </li></ul><ul><li>Record: RH 5-4. </li></ul><ul><li>Boys </li></ul><ul><li>Tigrett 52, Rose Hill 47 </li></ul><ul><li>Scoring: </li></ul><ul><li>T - Jaylen Barford 26, Malik Hicks 13, Bond 8, Cross 3, Love 2; </li></ul><ul><li>RH - Landon Simmons 13, Kendell Walker 11, Theus 7, Beauregard 8, Ferguson 4, Perkins 2, Beard 2. </li></ul><ul><li>3-pointers: RH - Simmons, Walker. </li></ul><ul><li>Halftime score: RH 19-18. </li></ul><ul><li>Record: RH 5-4. </li></ul>
11. 12. 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 6 gallons of gas. What is your fuel efficiency in miles per gallon? Rate = 120 miles ________ 6 gallons = ________ 20 miles 1 gallon Your fuel efficiency is 20 miles per gallon.
12. 13. Rate Ratio of two quantities measured in different units. A unit rate is a special rate where the denominator is 1. 60 miles per hour 20 miles for every gallon 60 minutes in one hour
13. 14. Unit Analysis Writing the units when comparing each unit of a rate is called unit analysis . Example: How many minutes are in 5 hours? 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. 5 hours • 60 minutes ________ 1 hour = 300 minutes
14. 15. Homework <ul><li>Page 272-273 </li></ul><ul><li>Ratios: 10, 11 </li></ul><ul><li>Unit Rate: 22, 23 </li></ul><ul><li>Equivalent Rate: 30, 31 </li></ul>
15. 16. Bell Ringer <ul><li>Given the following sets of points, give the slope of each line: </li></ul><ul><li>Line 1: (0,4) and (9,12) </li></ul><ul><li>Line 2: (-4,2) and (2, -3) </li></ul><ul><li>Line 3: (3, 5) and (3,8) </li></ul>
16. 17. Unit Analysis Writing the units when comparing each unit of a rate is called unit analysis . Example: How many minutes are in 5 hours? 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. 5 hours • 60 minutes ________ 1 hour = 300 minutes
17. 18. Bell Ringer <ul><li>Use a ‘conversion factor’ to solve the following conversion: </li></ul><ul><li>\$43 = ? Dollars </li></ul><ul><li>1 day 1 week </li></ul>
18. 19. Proportion An equation in which two ratios are equal is called a proportion . A proportion is written by setting two fractions EQUAL: a ___ ___ = b c d
19. 20. Read IT Correctly spell and pronounce the term Write IT Write the definition of the term in their own words Draw IT Represent the term through examples and visuals
20. 21. 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 ad=bc.
21. 22. Proportion Example: Write the original proportion. Use the reciprocal property. Multiply both sides by 35 to isolate the variable, then simplify.
22. 23. Proportion Example: Write the original proportion. Use the cross product property. Divide both sides by 6 to isolate the variable, then simplify.
23. 24. You Try It! If the average person lives for 75 years, how long would that be in seconds?
24. 25. 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.
25. 26. You Try It! John constructs a scale model of a building. He says that 3/4 th feet of height on the real building is 1/5 th 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?
26. 27. 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 3/4 th feet of height on the building… the model has 1/5 th 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.
27. 28. 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.
28. 29. Homework <ul><li>Page 283 </li></ul><ul><li>Comparing ratios to see if they are proportions: 12, 19 </li></ul><ul><li>Solving proportions: 20, 31 </li></ul><ul><li>Solving ‘two-step’ proportions: 37, 40 </li></ul><ul><li>Optional (for string): 41 </li></ul>