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Rates

Rates. Lesson 1. Real-World Link. ----Beats in 2 minutes = ? ----. Number of beats in 1 minute. Number of beats in 1 minute. -- minutes-- . -- minutes-- . Take your pulse for 2 minutes and record your results. Use your results to determine the number of beats for minute. .

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Rates

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  1. Rates Lesson 1

  2. Real-World Link ----Beats in 2 minutes = ? ---- Number of beats in 1 minute Number of beats in 1 minute --minutes-- --minutes-- Take your pulse for 2 minutes and record your results. Use your results to determine the number of beats for minute.

  3. Find a Unit Rate Rate: Definition: a ratio that compares two different quantities Example: Unit Rate: Definition: when a rate has a denominator of 1 Example:

  4. Common Unit Rates

  5. Example 1 Adrienne biked 24 miles in 4 hours. If she biked at a constant speed, how many miles did she ride in one hour? 24 miles in 4 hours = = Adrienne biked 6 miles in 1 hour.

  6. Got it? 1 Find each unit rate. Round to the nearest hundredth if necessary. a. $300 for 6 hours $50 per hour b. 220 miles on 8 gallons 27.5 miles per hour

  7. Example 2 Find the unit rate if it costs $2 for eight juice boxes. $2 for eight boxes = = The unit price is $0.25 per juice box.

  8. Example 3 40-pound bag: $49.00  40 ≈ $1.23 per pound 20-pound bag: $23.44  20 ≈ $1.17 per pound 8-pound bag: $9.88  8 ≈ $1.24 per pound The prices of 3 different bags of dog food are given in the table. Which size bag has the lowest price per pound rounded to the nearest cent?

  9. Got it? 3 Nutty: $0.18 per oz Grandma’s: $0.155 Bee’s: $0.1675 Sav-A-Lot: $0.165 Tito wants to buy some peanut butter to donate to the local food pantry. Tito wants to buy as much peanut butter as possible. Which brand should he buy?

  10. Example 4 Lexi painted 2 faces in 8 minutes at the Crafts Fair. At this rate, how many faces can she paint in 40 minutes? 2 faces in 8 minutes = = = x 40 min = 10 faces Lexi can paint 10 faces in 40 minutes.

  11. Complex Fractions and Unit Rates Lesson 2

  12. Complex Fractions Definition: Fractions when a numerator and/or denominator is also a fraction. Example:

  13. Example 1 Simplify . =  2 x =

  14. Example 2 Simplify . = 1  x =

  15. Example 3 = Josh jogs at an average speed of miles per hour. Josh can jog miles in hour. Find his average speed in miles per hour.

  16. Example 4 = Tia can paint 46 square feet per hour. Tia is painting her house. She paints square feet in hour. At this rate, how many square feet can she paint each hour?

  17. Got it? • Mr. Ito is spreading mulch in his yard. He spreads square yards in 2 hours. How many square yards can he mulch per hour? square yards per hour • Aubrey can walk miles in hours. Find her average speed in miles per hour. miles per hour

  18. Example 5 On Javier’s soccer team, about % of the players have scored a goal. Write % as a fraction in simplest form. % = =  100 = x =

  19. Convert Unit Rates Lesson 3

  20. Commonly Used Measurements

  21. Unit Ratio Like a unit rate, a unit ratio has a denominator of 1. Example:

  22. Example 1 Divide out the common units A remote control car travels at a rate of 10 feet per second. How many inches per second is this?

  23. Example 1 Simplify: So, 10 feet per second equals 120 inches per second. A remote control car travels at a rate of 10 feet per second. How many inches per second is this?

  24. Example 2 Divide out the common units A swordfish can swim at a rate of 60 miles per hour. How many feet per hour in this?

  25. Example 2 Simplify: Swordfish can swim at a rate of 316,800 feet per hour. A swordfish can swim at a rate of 60 miles per hour. How many feet per hour in this?

  26. Example 3 Divide out the common units Marvin walks at a speed of 7 feet per second. How many feet per hour is this?

  27. Example 3 Simplify: Marvin walks 25,200 feet in 1 hour. Marvin walks at a speed of 7 feet per second. How many feet per hour is this?

  28. Example 4 Divide out the common units The average speed of one team in a relay race is about 10 miles per hour. What is the speed in feet per second?

  29. Example 4 Simplify: The relay teams runs at an average speed of 14.7 feet per seconds The average speed of one team in a relay race is about 10 miles per hour. What is the speed in feet per second?

  30. Proportional and Nonproportional Relationships Lesson 4

  31. Proportional vs. Nonproportional Proportional – has a constant rate or a unit rate Nonproportional – does NOT have a constant rate or a unit rate These fractions are equivalent fractions because they all equal the same value.

  32. Example 1 Andrew earns $18 per hour for mowing lawns. Is the amount of money he earns proportional to the number of hours he spends mowing? Explain. Step 1: Make a table

  33. Example 1 Andrew earns $18 per hour for mowing lawns. Is the amount of money he earns proportional to the number of hours he spends mowing? Explain. Step 2: Make equivalent fractions Do they all equal each other? Yes, the amount Andrew earns is proportional to the number of hours he works.

  34. Got it? 1 At Lakeview Middle School, there are 2 homeroom teachers assigned to every 48 students. Is the number of students at this school proportional to the number of teachers? Explain your reasoning. The ratio is proportional since the ratio is 24 students to every teacher.

  35. Example 2 Uptown Tickets charges $7 per baseball game plus a $3 processing fee to order. Is the cost of an order proportional to the number of tickets ordered? Explain. STEP 1: Make a table.

  36. Example 2 STEP 2: Make equivalent fractions Are these fractions true? No, these are not equal so the cost and tickets ordered are not proportional.

  37. Example 3 You can use the recipe shown to make a fruit punch. Is the amount of sugar used proportional to the amount of mix used? Are the ratios equivalent? Yes, so the sugar and mix are proportional.

  38. Got it? 2 & 3 140 160 180 200 At the beginning of the year, Isabel had $120 in the bank. Each week, she deposits another $20. Is her account balance proportional to the number of weeks of deposits? Use the table below and explain your reasoning. No, the balance and the number of weeks are not proportion because the ratios are not equal.

  39. Example 4 The tables shown represent the number of pages Martin and Gabriel read over time. Which situation represents a proportional relationship? All of Martin’s ratios equal each other, so Martin’s table is proportional.

  40. MID-CHAPTER CHECK

  41. Graphing Proportional Relationships Lesson 5

  42. Identifying Proportional Relationships From a graph: A proportional relationship is… 1. a straight line 2. a line goes through the origin (0,0)

  43. Example 1 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. Step 1: Make a table

  44. Example 1 The slowest mammal on Earth is the tree sloth. It moves at a speed of 6 feet per minute. Determine whither the number of feet the sloth moves is proportional to the number of minutes it moves by graphing. Explain. Step 2: Graph the ordered pairs The line passes through the origin and the line is straight, so, this situation is proportional.

  45. Got it? 1 James earned $5 an hour babysitting. Determine whether the amount of money James earns babysitting is proportional to the number of hours he babysits by graphing. Explain. The amount of money earned is proportional to the number of hours because the line is straight and goes through the origin.

  46. Example 2 No, even though the line is straight, it does not go through the origin. The cost of renting video games from Games Inc. is shown in the table. Does this represent a proportional relationship? Explain.

  47. Got it? 2 No, even though the line goes through the origin, it is not a straight line. Determine is the number of calories and the number of minutes is proportional based on the table below.

  48. Example 3 Fun Center shows a proportional relationship because it goes through the origin. Which batting cage represents a proportional relationships between the number of pitches and the cost? Explain.

  49. Solve Proportional Relationships Lesson 6

  50. Write and Solve Proportions Definition: a proportion is an equation stating that two ratios or rates are equivalent. Numbers: Algebra:

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