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### 2 2 proportions

1. 1. Proportions Frank Ma © 2011
2. 2. Proportions Two related quantities stated side by side is called a ratio.
3. 3. Proportions Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4.
4. 4. 3 4 Proportions Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:
5. 5. 3 4 eggs cups of flour Proportions Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:
6. 6. 3 4 eggs cups of flour Proportions This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour. Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:
7. 7. 3 4 eggs cups of flour Proportions This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour. Two ratios that are equal are said to be in proportion. Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:
8. 8. 3 4 eggs cups of flour Proportions This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour. Two ratios that are equal are said to be in proportion. Thus "3 to 4" is proportion to "6 to 8" since 3 4 = 6 8 Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:
9. 9. 3 4 eggs cups of flour Proportions This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour. Two ratios that are equal are said to be in proportion. Thus "3 to 4" is proportion to "6 to 8" since 3 4 = 6 8 Proportional equations are the simplest type of fractional equations. Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:
10. 10. 3 4 eggs cups of flour Proportions This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour. Two ratios that are equal are said to be in proportion. Thus "3 to 4" is proportion to "6 to 8" since 3 4 = 6 8 Proportional equations are the simplest type of fractional equations. To solve proportional equations, we cross-multiply and change the proportions into regular equations. Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:
11. 11. A B C D , =If Cross-Multiplication-Rule Proportions
12. 12. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions
13. 13. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a.
14. 14. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply
15. 15. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x
16. 16. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x = x 5 6
17. 17. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x = x 2 3 (x + 2) (x – 5) =b. 5 6
18. 18. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x = x 5 6 2 3 (x + 2) (x – 5) =b. cross multiply
19. 19. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x = x 2 3 (x + 2) (x – 5) =b. cross multiply 2(x – 5) = 3(x + 2) 5 6
20. 20. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x = x 2 3 (x + 2) (x – 5) =b. cross multiply 2(x – 5) = 3(x + 2) 2x – 10 = 3x + 6 5 6
21. 21. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x = x 2 3 (x + 2) (x – 5) =b. cross multiply 2(x – 5) = 3(x + 2) 2x – 10 = 3x + 6 –10 – 6 = 3x – 2x 5 6
22. 22. A B C D , =If then AD = BC. Cross-Multiplication-Rule Proportions Example A. Solve for x. 3 x 5 2 =a. cross multiply 6 = 5x = x 2 3 (x + 2) (x – 5) =b. cross multiply 2(x – 5) = 3(x + 2) 2x – 10 = 3x + 6 –10 – 6 = 3x – 2x –16 = x 5 6
23. 23. Proportions When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions
24. 24. Proportions When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
25. 25. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
26. 26. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
27. 27. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
28. 28. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. number of eggs cups of flour When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
29. 29. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios. number of eggs cups of flour When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
30. 30. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios. number of eggs cups of flour x 10 When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
31. 31. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios. number of eggs cups of flour 3 4 x 10 When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
32. 32. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios. number of eggs cups of flour 3 4 x 10 = Set them equal, we get x 10 3 4 When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
33. 33. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios. number of eggs cups of flour 3 4 x 10 = Set them equal, we get x 10 3 4 cross multiply 4x = 30 When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
34. 34. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios. number of eggs cups of flour 3 4 x 10 = Set them equal, we get x 10 3 4 cross multiply 4x = 30 x = 30 4 When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
35. 35. Proportions Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios. number of eggs cups of flour 3 4 x 10 = Set them equal, we get x 10 3 4 cross multiply 4x = 30 x = 30 4 = 7½ We need 7½ eggs. When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.
36. 36. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions
37. 37. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as
38. 38. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as miles inches
39. 39. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as miles inches x 14
40. 40. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as miles inches 21 4 x 14
41. 41. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as miles inches 21 4 x 14 Set them equal, we get 21 4 x 14 =
42. 42. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as miles inches 21 4 x 14 Set them equal, we get 21 4 x 14 = cross multiply 4x = 294
43. 43. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as miles inches 21 4 x 14 Set them equal, we get 21 4 x 14 = cross multiply 4x = 294 x = 294 4
44. 44. Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map? Proportions Let x be the number of real distance in miles and write the two types of measurements as miles inches 21 4 x 14 Set them equal, we get 21 4 x 14 = cross multiply 4x = 294 x = 294 4 = 73½ So 4 inches corresponds to 73½ miles.
45. 45. Proportions Geometric proportion gives us the sense of “similarity”, that is, objects that are of the same shape but different sizes.
46. 46. Proportions Given the following cats, which cat on the right is of the same shape as the one on the left? A. B. Geometric proportion gives us the sense of “similarity”, that is, objects that are of the same shape but different sizes.
47. 47. Proportions Given the following cats, which cat on the right is of the same shape as the one on the left? The answer of course is A. A. B. Geometric proportion gives us the sense of “similarity”, that is, objects that are of the same shape but different sizes.
48. 48. Proportions Given the following cats, which cat on the right is of the same shape as the one on the left? The answer of course is A. A. B. This sense of similarity is due to the fact that “all corresponding linear measurements are in proportion”. Geometric proportion gives us the sense of “similarity”, that is, objects that are of the same shape but different sizes.
49. 49. Proportions For example, given the following similar cats,
50. 50. Proportions For example, given the following similar cats, let’s identify some corresponding points as shown.
51. 51. Proportions B. For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. 6 10 12
52. 52. Proportions B. For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. 6 10 12 3
53. 53. Proportions B. For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. 6 10 12 3 y z
54. 54. Proportions B. For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio, 6 10 12 3 y z
55. 55. Proportions B. For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio, we see that y = 6 10 12 3 y z
56. 56. Proportions B. For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio, we see that y = 10/5 = 2 and z = 12/2 = 6. 6 10 12 3 y z
57. 57. Proportions 6 10 12 3 5 6 For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio, we see that y = 10/5 = 2 and z = 12/2 = 6.
58. 58. Proportions Definition. Two geometric objects are similar if all corresponding linear measurements are in proportion. B.
59. 59. Proportions Definition. Two geometric objects are similar if all corresponding linear measurements are in proportion. B. Hence if X, Y and Z are the measurements from the original cat as shown, X Y Z
60. 60. Proportions Definition. Two geometric objects are similar if all corresponding linear measurements are in proportion. B. Hence if X, Y and Z are the measurements from the original cat as shown, and x, y and z are the corresponding measurements in the similar copy, X Y Z x y z
61. 61. Proportions Definition. Two geometric objects are similar if all corresponding linear measurements are in proportion. B. Hence if X, Y and Z are the measurements from the original cat as shown, and x, y and z are the corresponding measurements in the similar copy, then it must be that X Y Z x y z X x = Y y = Z z
62. 62. Proportions Definition. Two geometric objects are similar if all corresponding linear measurements are in proportion. B. Hence if X, Y and Z are the measurements from the original cat as shown, and x, y and z are the corresponding measurements in the similar copy, then it must be that X Y Z x y z X x = Y y = Z z or that X Y = x y etc…
63. 63. Proportions Definition. Two geometric objects are similar if all corresponding linear measurements are in proportion. B. Hence if X, Y and Z are the measurements from the original cat as shown, and x, y and z are the corresponding measurements in the similar copy, then it must be that X Y Z x y z X x = Y y = Z z or that X Y = x y etc… This mathematical formulation of similarity is the basis for biometric security software such as facial recognition systems.
64. 64. Proportions The reason that linear proportionality gives us the sense of similarity is due to visual geometry, that is, the visual images of an object at different distances form a “cone”.
65. 65. Proportions The reason that linear proportionality gives us the sense of similarity is due to visual geometry, that is, the visual images of an object at different distances form a “cone”.
66. 66. Proportions The reason that linear proportionality gives us the sense of similarity is due to visual geometry, that is, the visual images of an object at different distances form a “cone”. One can easily demonstrate that in such projection, similar images of two different sizes must preserve the ratio of two corresponding distance measurements – as in the above definition of similarity.
67. 67. Proportions The reason that linear proportionality gives us the sense of similarity is due to visual geometry, that is, the visual images of an object at different distances form a “cone”. One can easily demonstrate that in such projection, similar images of two different sizes must preserve the ratio of two corresponding distance measurements – as in the above definition of similarity. Similar triangles
68. 68. Proportions In the definition of similarity, the measurements in proportion must be linear.
69. 69. Proportions In the definition of similarity, the measurements in proportion must be linear. It's not true that the ratio of areas is proportional to linear ratio of two similar objects.
70. 70. Proportions In the definition of similarity, the measurements in proportion must be linear. It's not true that the ratio of areas is proportional to linear ratio of two similar objects. Because the ratio of the linear measurements is 1 : 2, 1 2
71. 71. Proportions In the definition of similarity, the measurements in proportion must be linear. It's not true that the ratio of areas is proportional to linear ratio of two similar objects. Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the area–measurements is 1 : (2)2 or 1 : 4 ratio. 1 2 linear ratio 1 : 2 area–ratio 1 : 4 = 22 :
72. 72. Proportions 1 2 linear ratio 1 : 2 area–ratio 1 : 4 = 22 : In the definition of similarity, the measurements in proportion must be linear. It's not true that the ratio of areas is proportional to linear ratio of two similar objects. Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the area–measurements is 1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to paint the larger one.
73. 73. Proportions 1 2 linear ratio 1 : 2 area–ratio 1 : 4 = 22 linear ratio 1 : r area–ratio 1 : r2 1 r : : In the definition of similarity, the measurements in proportion must be linear. It's not true that the ratio of areas is proportional to linear ratio of two similar objects. Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the area–measurements is 1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to paint the larger one. Given two similar objects with linear ratio 1 : r then their area–ratio is 1 : r2.
74. 74. Proportions 1 2 linear ratio 1 : 2 area–ratio 1 : 4 = 22 linear ratio 1 : r area–ratio 1 : r2 1 r : : In the definition of similarity, the measurements in proportion must be linear. It's not true that the ratio of areas is proportional to linear ratio of two similar objects. Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the area–measurements is 1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to paint the larger one. Given two similar objects with linear ratio 1 : r then their area–ratio is 1 : r2. This is also true for the surface areas.
75. 75. Proportions 1 2 linear ratio 1 : 2 area–ratio 1 : 4 = 22 linear ratio 1 : r area–ratio 1 : r2 1 r : : In the definition of similarity, the measurements in proportion must be linear. It's not true that the ratio of areas is proportional to linear ratio of two similar objects. Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the area–measurements is 1 : (2)2 or 1 : 4 ratio. So it would take 4 times as much paint to paint the larger one. Given two similar objects with linear ratio 1 : r then their area–ratio is 1 : r2. This is also true for the surface areas i.e. two similar 3D solids of 1 : r linear ratio have 1 : r2 as their surface–area ratio.
76. 76. Proportions The volume of an object is the measurement of the 3-dimensional space the object occupies.
77. 77. Proportions Because the ratio of the linear measurements is 1 : 2, : linear ratio 1 : 2 1 2 The volume of an object is the measurement of the 3-dimensional space the object occupies.
78. 78. Proportions Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the volume–measurements is 1 : (2)3 or 1 : 8 ratio. : linear ratio 1 : 2 volume–ratio 1 : 8 = 23 1 2 The volume of an object is the measurement of the 3-dimensional space the object occupies.
79. 79. Proportions Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the volume–measurements is 1 : (2)3 or 1 : 8 ratio. So the weight of large one is 8 times as much as the small one (if they are made of the same thing).* : linear ratio 1 : 2 volume–ratio 1 : 8 = 23 1 2 The volume of an object is the measurement of the 3-dimensional space the object occupies.
80. 80. Proportions Because the ratio of the linear measurements is 1 : 2, we may conclude that the ratio of the volume–measurements is 1 : (2)3 or 1 : 8 ratio. So the weight of large one is 8 times as much as the small one (if they are made of the same thing).* Given two similar objects with linear ratio 1 : r then their volume–ratio is 1 : r3. linear ratio 1 : r volume–ratio 1 : r3 :: linear ratio 1 : 2 volume–ratio 1 : 8 = 23 1 2 1 r The volume of an object is the measurement of the 3-dimensional space the object occupies.
81. 81. The Golden Ratio Proportions
82. 82. The Golden Ratio Proportions An important ratio in arts and science is the golden ratio.
83. 83. The Golden Ratio Proportions An important ratio in arts and science is the golden ratio. Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”,
84. 84. The Golden Ratio Proportions An important ratio in arts and science is the golden ratio. Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1
85. 85. The Golden Ratio Proportions Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L An important ratio in arts and science is the golden ratio.
86. 86. The Golden Ratio Proportions Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L An important ratio in arts and science is the golden ratio. 1 L
87. 87. The Golden Ratio Proportions Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L s An important ratio in arts and science is the golden ratio. L s 1 L =
88. 88. The Golden Ratio Proportions Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L s An important ratio in arts and science is the golden ratio. L s 1 L = The ratio1/L is the Golden Ratio.
89. 89. The Golden Ratio Proportions L (1 – L) 1 L Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L s An important ratio in arts and science is the golden ratio. L s 1 L = Since s = 1 – L, we’ve = The ratio1/L is the Golden Ratio.
90. 90. The Golden Ratio Proportions L (1 – L) 1 L cross multiply Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L s An important ratio in arts and science is the golden ratio. L s 1 L = Since s = 1 – L, we’ve = 1 – L = L2 The ratio1/L is the Golden Ratio.
91. 91. The Golden Ratio Proportions L (1 – L) 1 L cross multiply 0 = L2 + L – 1 Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L s An important ratio in arts and science is the golden ratio. L s 1 L = Since s = 1 – L, we’ve = 1 – L = L2 The ratio1/L is the Golden Ratio.
92. 92. The Golden Ratio Proportions L (1 – L) 1 L cross multiply 0 = L2 + L – 1 Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1 L s An important ratio in arts and science is the golden ratio. L s 1 L = Since s = 1 – L, we’ve = 1 – L = L2 We’ll see later that the answer for L is about 0.618. http://en.wikipedia.org/wiki/Golden_ratio http://en.wikipedia.org/wiki/Golden_spiral The ratio1/L is the Golden Ratio.
93. 93. Ex. A. Solve for x. Proportions 3 2 2x + 1 3 = 21 x 3 =1. –2x/7 –5 1½ 3 =9. 2. 6 2=3. x + 1 –5 7 =4. –2x + 1 6 2=5. x + 1 2 3x =6. 5x + 1 3 4x = 8 2/3 7/5 x 14 =7. 9 4/3 3 2x/5 =8. 10. – x + 2 2 –4 3 =11. 3 2x + 1 1 x – 2 =12. x – 5 2 2x – 3 3 =13. 3 2 x – 4 x – 1 =14. –3 5 2x + 1 3x =15. 3x + 2 2 2x + 1 3 =16. 3 2 2x + 1 x – 3 =17. 2 x x + 1 3 =18. x 2 x + 4 x =19.
94. 94. Ex. B. (Solve each problem. It’s easier if fractional proportions are rewritten as proportions of integers.) Proportions 20. If 4 cups of flour need 3 tsp of salt, then 10 cups flour need how much of salt? Different cookie recipes are given, find the missing amounts. 21. If 5 cups of flour need 7 cups of water then 10 cups water need how much of flour? 23. If 2 ½ tsp of butter flour need 3/4 tsp of salt, then 6 tsp of salt need how much of flour. 22. If 5 cups of flour need 7 cups of water then 10 cups flour need how much of water? 24. If 1¼ inches equals 5 miles real distance, how many miles is 5 inches on the map? For the given map scales below, find the missing amounts. 25. If 2½ inches equals 140 miles real distance, how many inches on the map correspond to the distance of 1,100 miles?
95. 95. Proportions A . B . C . 3 4 1 5 5 6 26. x 20 x 20 6 27. x 3x + 4 2xx+2 15 x x 2x – 3 28. 29. 30. 31. 32–37. Find the surface areas of each cat above if A’s surface area is 2 ft2, B’s surfaces area is 7 ft2. and C’s is 280 ft2? Ex. C. 26 – 31. Solve for x. Use the given A, B and C,. 38–43. Find the weight of each cat above if cat A is 20 lbs, cat B is 8 lb. and cat C is 350 lb?