Ratio, Rates, & Proportions

1. Ratio, Rates, & Proportions

2. Ratios • A ratio is a comparison of two numbers. • Example: Tamara has 2 dogs and 8 fish. The ratio of dogs to fish can be written in three different ways. *** Be careful with the fraction ratios – they don’t always have identical meanings to other fractions***

3. There are two different types of ratios • .Part-to-Part Ratios • Example: Tamara’s dogs to cats is 2 to 8 or 1 to 4 • .Part –to-Whole Ratios • Example: Tamara’s dogs to total pets is 2 to 10 or 1 to 5

4. Try the following Apples to bananas Oranges to apples There are 15 oranges and 12 apples So the ratio of oranges to apples is 5:4, 5 to 4, OR • There are 12 apples and 3 bananas • So the ratio of apples to bananas is 4:1, 4 to 1, OR

5. A large bouquet of flowers is made up of 18 roses, 16 daisies, and 24 irises. Write each ratio in all three forms in simplest form. Identify which ratios are part-to-part and which ratios are part-to-whole. Iris to daisies all flowers to roses • Roses to iris • Daisies to roses 18 to 24 3 to 4 24 to 16 3 to 2 16 to 18 8 to 9 58 to 8 29 to 9

6. 9 Explain why the ratio • The ratio is comparing two number. If you changed it to a mixed number it would no longer be a comparison. This is why fraction ratios are tricky  is not written as a mixed number:

7. Comparing Ratios • Compare ratios by writing in simplest form • Are these ratios equivalent? 250 Kit Kats to 4 M&M’s and 500 Kit Kats to 8 M&M’s • 12 out of 20 doctors agree and 12 out of 30 doctors agree • The ratio of students in Ms. B’s classes that had HW was 8 to 2 and 80% of Mrs. Long’s class had HW = ≠ =

8. Rates • A rate is a ratio that compares two different quantities or measurements. • Rates can be simplified • Rates us the words per and for • Example: Driving 55 miles per hour • Example: 3 tickets for $1

9. Unit Rates • A unit rate is a rate per one unit. In unit rate the denominator is always one. • Example: Miguel types 180 words in 4 min. How many words can he type per minute? 45 . 1 45 or words per minute = rate unit rate word form

10. Unit rates make it easier to make comparisons. • Example: Taylor can type 215 words in 5 min • How many words can he type per minute? • Who is the faster typist? How much faster? Taylor is 2wpm faster than Miguel

11. Try the Following Drive 288 miles on 16 gallons of gas. Earn $49 for 40 hours of work Use 5 ½ quarts of water for every 2 lbs of chicken Sarah drove 5 miles in 20 minutes $2.50 per roll 18 mpg • Film costs $7.50 for 3 rolls • 90 students and 5 teachers • **Snowfall of 12 ¾ inches in 4 ½ hours. • Ian drove 30 miles in 0.5 hours 18 students per teacher $1.23 per hour 2 5/6 per hour 2 ¾ quarts per lb 60 mph 15 mph

12. Complex Unit Rates • Suppose a boat travels 30 miles in 2 hours • How do you write this rate? • Suppose a boat travels 12 miles in 2/3 hours • How do you write this as a rate? • How do you write this as a division problem? • Determine the unit rate: 18mph

13. Suppose a boat travels 8 ¾ miles in 5/8 hours. • How do you write this as a rate? • How do you write this as a division problem? • Determine the unit rate: 14 mph

14. .Complex fractions are fractions that have fractions within them. They are either in the numerator, denominator, or both. • Divide complex fractions by multiplying (keep, change, change)

15. TRY the FOLLOWINGWrite each rate. Then determine the unit rate and write in both fraction and word form Mrs. Robare is making costumes for the school play. Each costume requires 0.75 yards of fabric. She bought 6 yards of fabric. How many costumes can Mrs. Robare make? A lawn company advertises that they can spread 7,500 square feet of grass seed in hours. Find the number of square feet of grass seed that can be spread in an hour. • Mary is making pillows for her Life Skills class. She bought yards of fabric. Her total cost was $16. What was the cost per yard? • Doug entered a canoe race. He rowed miles in hour. What is his average speed? $5.82 per yard 8 costumes 7mph 3000 ft per hour

16. Comparing Unit RatesDario has two options for buying boxes of pasta. At CornerMarket he can buy seven boxes of pasta for $6. At SuperFoodz he can buy six boxes of pasta for $5. • He divided 7 by 6 and got 1.166666667 at CornerMarket. He then divided 6 by 7 and got 0.85714286. He was confused. What do these numbers tell him about the price of boxes of pasta at CornerMarket? • Decide which makes more sense to you • Compare the two stores’ prices. Which store offers the better deal? 1.166667 is the number of boxes you can get for $1 0.85714286 is the price per box Price per box CM - $0.86 per box SF - $0.83 per box

17. Proportions

18. Two quantities are proportional if they have a constant ratio or unit rate. • You can determine proportionality by comparing ratios • Andrew earns $18 per hour for mowing lawns. Is the amount he earns proportional to the number of hours he spends mowing? • Make a table to show these amounts • For each number of hours worked, write the relationship of the amount he earned and hour as a ratio in simplest form. • Are all the rates equivalent? Since each rate simplifies to 18, they are all equivalent. This means the amount of money Andrew earns is proportional to the number of hours he spends mowing.

19. Uptown Tickets charges $7 per baseball game ticket plus $2 processing fee per order. Is the cost of an order proportional to the number of tickets ordered? • Make a table to show these amounts • For each number of tickets, write the relationship of the cost of the and the number of tickets ordered. • Are all the rates equivalent? The rates are not equivalent. This means the total cost of the tickets is not proportional to the number of tickets sold.

20. Use the recipe to make fruit punch. Is the amount of sugar used proportional to the amount of mix used? Explain. • Yes, they all reduce to ½ • In July, a paleontologist found 368 fossils at a dig. In August, she found about 14 fossils per day. • Is the number of fossils the paleontologist found in August proportional to the number of days she spent looking for fossils that month? No, July average 11.87 fossils per day

21. Solving Proportions

22. A proportion is two equivalent ratios • When solving proportions we must first ask ourselves – “What are we comparing A lemonade recipe calls for ½ cup of mix for every quart of water. If Jeff wanted to make a gallon of lemonade, is 2 cups of mix proportional for this recipe? YES Determine if the following ratios are proportional? No, 340 ≠ 270 Yes, 72 = 72

23. b. c. • Proportionality can also be determined between two ratios by simplifying or comparing their cross products • If they reduce to the same ratio, or their cross products are the same, then they are proportional • You can also solve proportions for a missing variable by cross multiplying. • Example: Determine if the two ratios are proportional: Yes 72 = 72 No 30 = 70 No 32 = 30

24. x = 60 • Example: Determine the value of x: • Example: A stack of 2,450 one-dollar bills weighs five pounds. How much do 1,470 one-dollar bills weigh? • Set up a proportion – ask ourselves “what are we comparing?” • Example: Whitney earns $206.25 for 25 hours of work. At this rate, how much will Whitney earn for 30 hours of work? • How much does Whitney earn per hour? 3 pounds $247.50 $8.25 per hour

25. Coordinate Plane Review

26. The Coordinate Plane Quadrant II Quadrant I x-axis y-axis Origin Quadrant III Quadrant IV

27. Ordered Pair: is a pair of numbers that can be used to locate a point on a coordinate plane

28. Ordered Pairs • Ordered Pair: is a pair of numbers that can be used to locate a point on a coordinate plane. • Example: (3, 2) • y - coordinate II • x - coordinate I ● III IV ●

29. Graph the following ordered pairs on the coordinate plan and state the quadrants the points are located in • (3, 2) • (-5, 4) • (6, -4) • (-7, 7) I ● II ● IV ● III ●

30. Steps for Graphing

31. Draw and label the x and y axis – don’t forget your arrows • Make a table of values to represent the problem. Be sure to include the values: 0, 1, and 2 • Graph your order pairs- you need at least 3 points to make a line • Draw a line through the points – don’t forget your arrows • If the line is straight and goes through the origin, then the quantities are proportional

32. Example:The slowest mammal on Earth is the tree sloth. It moves at a speed of 6 feet per minute. Determine whether the number of feet the sloth moves is proportional to the number of minutes it moves by graphing. Explain your reasoning. y Yes – it is a straight line through the origin x

33. Example:The table below shows the number of calories an athlete burned per minute of exercise. Determine whether the number of calories burned is proportional to the number of minutes by graphing. Explain your reasoning. y No – it is not a straight line and it doesn’t go through the origin x

34. Slope

35. .Slope is the rate of change between any two points on a line • The sign of the slope tells you whether the line is positive or negative. • You can find slope of a line by comparing any two points on that line • Slope is the or

36. The line goes down 2 (rise) and over 1 (run). • Slope = -2 Positive Slope Negative Slope • The line goes up 3 (rise) and over 1 (run). • Slope = 3

37. Tell whether the slope is positive or negative . Then find the slope • Negative • Slope = -1 • Positive • Slope = 4/3

38. Use the given slope and point to graph each line

39. Use the given slope and point to graph each line

40. Use the given slope and point to graph each line

41. Rate of Change (Slope)

42. .Rate of change (slope) describes how one quantity changes in relation to another. For graphs, the rate of change (slope) is constant ( a straight line)

43. Tell whether each graph shows a constant or variable rate of change Constant Variable

44. Tell whether each graph shows a constant or variable rate of change Variable Constant

45. Tell whether each graph shows a constant or variable rate of change Constant Variable

46. Proportional Relationships

47. A proportional relationship between two quantities is one in which the two quantities vary directly with one another (change the same way). This is called a direct variation. For graphs, the rate of change (slope) is constant ( a straight line)

48. Determine whether each table represents a direct variation by comparing the ratios to check for a common ratio This table represents a direct variation and is proportional This table does NOT represent a direct variation and is NOT proportional

49. Determine whether each table represents a direct variation by comparing the ratios to check for a common ratio This table does NOT represent a direct variation and is NOT proportional This table represents a direct variation and is proportional

50. The equations of such relationships are always in the form y = mx and when graphed produce a line that passes through the origin.