Proportions!!!

1. Proportions!!! Solving simple ones Notes for January 13

2. Word of the Day Inane stupid; dumb; pathetic

3. Today’s Objective • IWBAT solve algebraic proportions.

4. 45 120 14 16 9 72 24 64 1. 3. 2. 4. WARM-UP Write each fraction in lowest terms (simplify). 7 8 3 8 1 8 3 8

5. A ratiois a comparison of two quantities by division. Ratios that make the same comparison are equivalent ratios. In one rectangle, the ratio of shaded squares to unshaded squares is 7:5. In the other rectangle, the ratio is 28:20. Both rectangles have equivalent shaded areas. 28:20 7:5

6. 9 27 9 • 2 27 • 2 = = Two ratios equivalent to are and . Two ratios equivalent to are and . 9 27 9 ÷ 9 27 ÷ 9 64 24 9 27 128 48 18 54 1 3 8 3 = = 64 24 64 24 64 ÷8 24 ÷ 8 64 • 2 24 • 2 = = = = Example 1: Finding Equivalent Ratios Find two ratios that are equivalent to each given ratio. Multiply or divide the numerator and denominator by the same nonzero number. 18 54 A. 1 3 128 48 B. 8 3

7. Ratios that are equivalent are said to be proportional, or in proportion. Equivalent ratios are identical when they are written in simplest form.

8. 1 9 1 9 12 15 3 27 27 36 2 18 Since , the ratios are in proportion. B. A. = and and 2 18 3 27 3 ÷ 3 27 ÷ 3 2 ÷ 2 18 ÷ 2 = = = = 4 5 3 4 Since , the ratios are not in proportion.  12 15 27 36 27 ÷ 9 36 ÷ 9 12 ÷ 3 15 ÷ 3 = = = = Simplify to tell whether the ratios form a proportion. 1 9 1 9 4 5 3 4

9. 1 5 1 5 14 49 3 15 9 45 16 36 Since , the ratios are in proportion. B. A. = and and 3 15 9 45 3 ÷ 3 15 ÷ 3 9 ÷ 9 45 ÷ 9 = = = = 2 7 4 9 Since , the ratios are not in proportion.  14 49 16 36 16 ÷ 4 36 ÷ 4 14 ÷ 7 49 ÷ 7 = = = = Simplify to tell whether the ratios form a proportion. 1 5 1 5 2 7 4 9

10. We can also use cross products to figure out whether two ratios are in proportion.

11. 6 15 6 15 4 10 4 10 ? = Tell whether the ratios are proportional. 60 Find cross products. 60 60 = 60 Since the cross products are equal, the ratios are proportional.

12. Algebraic Proportions • Algebraic proportions are the same as regular proportions. • The cross-products must equal each other! KEYPOINT

13. Solving Algebraic Proportions • To solve algebraic proportions, follow these steps: 1.) Cross-multiply 2.) Set the products equal to each other 3.) Solve for x 4.) Box your answer

14. Solving Algebraic Proportions • The most important thing to remember is to: Kris Kross

15. Solving Algebraic Proportions • Solve for x in the following proportion:

16. Solving Algebraic Proportions 2(12) = 24 • Cross-multiply 4(x) = 4x

17. Solving Algebraic Proportions • Set the products equal to each other 4x = 24 What am I called?

18. Solving Algebraic Proportions • Solve for x

19. Solving Algebraic Proportions • Solve for x in the following proportion:

20. Solving Algebraic Proportions 5(-6) = -30 • Cross-multiply x(15) = 15x

21. Solving Algebraic Proportions • Set the products equal to each other 15x = -30 What am I called?

22. Solving Algebraic Proportions • Solve for x

23. Try some with your partner!

24. Try some on your own!

25. Proportions!!! Solving complex ones Notes for January 14th

26. Let’s not make it too hard to begin with. Let’s start by just throwing a coefficient in front of the x.

27. More Complex Algebraic Proportions • What happens when you see one of these? DO THE SAME THING!!!

28. More Complex Algebraic Proportions • Cross-multiply 2x(10) = 20x 8(5) = 40

29. More Complex Algebraic Proportions • Set the products equal to each other 20x = 40 What am I called?

30. More Complex Algebraic Proportions • Solve for x

31. More Complex Algebraic Proportions • Solve the following proportion

32. More Complex Algebraic Proportions • Cross-multiply 20(3x) = 60x 12(5) = 60

33. More Complex Algebraic Proportions • Set the products equal to each other 60x = 60 What am I called?

34. More Complex Algebraic Proportions • Solve for x

35. Try some with your partner!

36. As a kicker, I have much expertise in this manner … LET’S KICK IT UP!!!

37. Even more complex algebraic proportions! • What happens when you see a proportion?

38. KEYPOINT!!! • When solving proportions like that, you must remember that each numerator and denominator are together – like a couple. You cannot separate them. So in order to do this, you must use the Distributive Property.

39. Steps for Solving Complex Proportions 1.) Cross-Multiply 2.) Set the products equal to each other 3.) Use the Distributive Property 4.) Solve for x 5.) Box your answer

40. Even more complex algebraic proportions • Cross-multiply 2(x – 2) = 2(x – 2) -4(5) = -20

41. Even more complex algebraic proportions • Set the products each to each other 2(x – 2) = -20

42. Even more complex algebraic proportions • Use the Distributive Property and solve for x

43. Even more complex algebraic proportions! • Solve the following proportion:

44. Even more complex algebraic proportions -2(x + 6) = -2(x + 6) • Cross-multiply 3(x - 5) = 3(x – 5)

45. Even more complex algebraic proportions • Set the products each to each other -2(x + 6) = 3(x – 5)

46. Even more complex algebraic proportions • Use the Distributive Property and solve for x

47. On Your Own!

48. PRACTICE! It’ll be a Party in Ms. Ryan’s Room!