1. Ratios Rates Proportions

2. A Ratio is a comparison of two quantities by division. Equivalent Ratios: two ratios that name the same number.

3. A rate is a ratio that compares two quantities measured in different ways. Unit rate is the rate for one unit of a given quantity. Unit cost is a unit rate that gives the cost per unit.

4. A proportion is an equation stating that 2 ratios are equal. The cross products are the two products found by multiplying the denominator of each ratio by the numerator of the other ratio.

5. Ratio: 3:1, 3/1, 3 to 1, 1:3, 1/3, 1 to 3, etc… Conversions Factors & Rates: 55mph, 25mpg, 5ft/s, etc… Proportions: 3/1 and 6/2 2/3 and 6/9 12/16 and 15/20

6. Answer True or False on each of the following Pair of ratios if they form a proportion.

7. True or False: ― = ― 4242 20 6

8. Answer: False

9. True or False: ― = ― 3232 18 8

10. Answer: False

11. True or False: ― = ― 4343 16 12

12. Answer: True

13. True or False: ― = ― 4343 8686

14. Answer: True

15. True or False: ― = ― 12 24 3434

16. Answer: False

17. True or False: ― = ― 6969 2323

18. Answer: True

19. Solve the Following

20. Solve: ― = ― m 10 3

21. Answer: m = 33⅓

22. Solve: ― = ― v 12 10 2

23. Answer: v = 60

24. Solve: ― = ― 10 k 8484

25. Answer: k = 5

26. Solve: ― = ― 2x2x 7979

27. Answer: x = 2 ⁴⁄₇

28. Solve: ― = ― 3x3x 7 10

29. Answer: x = 4 ²⁄₇

30. Ratio & Proportions Parts = a & b Whole = (a+b) Part(a):Part(b) = a:b Proportions a:b = x·a:x·b Part:Whole = a:(a+b) = b:(a+b)

31. The ratio of kids to adults is 5 to 2. If there is 14 adults, how many kids are there?

32. Kids:adults=5:2 adults=14 Lets try… 5:2 = 2·5:2·2 = 10:4 (no) 3·5:3·2 = 15:6 (no) 4·5:4·2 = 20:8 (no)… 7·5:7·2 = 35:14 35kids:14adults

33. Kids:adults=5:2 kids=k adults=14 Or lets try… 5/2 = k/14 (5/2)·14 = k 35 = k Answer: 35 kids

34. Kids:adults=5:2 adults=14 Or lets try… Find the factor: 5x kids 2x adults 2x = 14 x = 7 (factor) Therefore: 5x (kids) 5·7 =35 Answer: 35 kids

35. The ratio of kids to adults is 3 to 5. If there are 26 more adults than kids, how many kids are there?

36. Kids:adults=3:5 26 more adults than kids Lets try… 3:5 = 10·3:10·5 = 30:50 (no) 15·3:15·5 = 45:75 (no) 13·3:13·5 = 39:65 39kids:65adults

37. Kids:adults=3:5 Find the factor: 3x kids & 5x adults adults – kids=26 5x-3x=26 2x=26 x = 13 (factor) Therefore: 3x (kids) 3·13 =39 Answer: 39 kids

38. Kids:adults=3:5 k kids k+26 adults 3/5 = k/(k+26) 5k = 3(k+26) 5k = 3k + 78 2k = 78 k = 39 Answer: 39 kids

39. 30 boys and girls are planning for a picnic. There are a ratio of 3 girls to 7 boys. How many boys are there?

40. Answer: 30 ∙ ⁷⁄₁₀ = 21 3x + 7x = 30 10x = 30 x=3 3·3 3·7 9/21 (girls/boys)

41. I have 70 friends. Among them are 11 girls for every 3 boys. How many friends are boys? How many are girls?

42. Answer: 70 ∙ ¹¹⁄₁₄ = 55 girls 70 ∙ ³⁄₁₄ = 15 boys Also: find the factor 11x + 3x = 70 14x=70 x=5 11·5=55 girls 3·5=15 boys

43. The ratio of dogs to cats is 9 to 2. If there are 72 dogs, how many total dogs and cats are there?

44. Answer: 88

45. To get a shade of orange paint, we mix white, red, & yellow in the ratio 2:5:4. If we need 6 gallons of orange paint, then how many gallons of each color do we need?

46. Answer: White=2x, red=5x, yellow=4x 2x+5x+4x=6 X=6/11 White=12/11, red=30/11, yellow=24/11

47. The top 3 cash prizes in the math tournament are in the ratio 5:2:1. The 1 st place prize is $20,000 more than the 3 rd place prize. How much is the second place prize?

48. Answer: 1st=5x, 2nd=2x, 3rd=1x 5x-1x=20000 X=5000 2 nd place = 2x=2∙5000=$10,000

49. More Proportion Word Problems

50. 5 pizzas cost $60. What will 9 pizzas cost?

51. Answer: ⁶⁰⁄₅ ∙ 9 = 108

52. 6 stuffed peppers cost $36. What will 12 stuffed peppers cost?

53. Answer: ³⁶⁄₆ ∙ 12 = 72

54. 10 glasses cost $40. How much do 20 glasses cost?

55. Answer: ⁴⁰⁄₁₀ ∙ 20 = $80

56. Twenty-eight is to five as eight hundred ninety-six is to _?_.

57. Answer: ― = ― x = 160 28 5 896 x

58. If you can buy one can of pineapple chunks for $2, then how many can you buy with $10?

59. Answer: ― = ― x = 5 cans 1212 x 10

60. One jar of crushed ginger cost $2. How many jars can you buy for $4?

61. Answer: ― = ― x = 2 jars 1212 x4x4

62. Lauren divided her money in the ratio 4:2 between Haley and Makenzie. Haley got the smaller amount of $1256. How much did Makenzie receive?

63. ²⁄₆ ∙ T = 1256 T=1256 ∙ ⁶⁄₂ Total= 3768 Makenzie received 3768 – 1256 = $2512 or 4/2 = M/1256 2M = 4·1256 M =2512

64. 12 miles is approximately equal to 6 km. How many miles are equal to 42 km?

65. Answer: ¹²⁄₆ ∙ 42 = 84

66. 12 miles is approximately equal to 6 km. How many km are equal to 18 miles?

67. Answer: ⁶⁄₁₂ ∙ 18 = 9

68. Conversion Word Problems

69. How many inches are equal to 8 yards?

70. Answer: 8 yds ∙ 3ft/1yd ·12in/1ft = 288 inches

71. How many inches are equal to 2 miles?

72. Answer: 2 miles ∙ 5280ft/1mi ·12in/1ft = 126,720 inches

73. How many seconds are equal to 2 years?

74. Answer: 2 yr∙(365days/1yr)·(24hr/1day)·(60min/1hr)· (60sec/1min)= 63,072,000 seconds

75. How many cups are in 5 gallons?

76. Answer: 5 gal∙(4qt/1gal)·(2pint/1qt)·(2cups/1pint)= 80 cups

77. I am making some math videos. I shoot 10 “takes” for each video and expect to make 144 videos per book. I have 11 books. Each “take” averages 5 minutes. If I shoot continuously, how many days will it take me to make a full set of videos for all the books?

78. Answer: 11bk∙(144vid/1bk)·(10takes/1vid)·(5min/1take)· (1hr/60min)·(1day/24hr) 55 days

79. Similar Figures

80. Polygon is a closed plane figure formed by 3 or more line segments that do not cross. Two polygons are similar polygons if: corresponding angles have the same measure. the lengths of the corresponding sides form equivalent ratios.

81. Example: BC/GH = AC/FH (Text Book) 50/75 = 40/60 or BC/AC = GH/FH 50/40 = 75/60

82. Each pair of figures are similar. Find the missing side. 20x 12 3

83. Answer: ― = ― x = 5 x 20 3 12

84. Each pair of figures are similar. Find the missing side. 4 x 16 8

85. Answer: ― = ― x = 2 x4x4 8 16

86. Each pair of figures are similar. Find the missing side. 14 x1 2

87. Answer: ― = ― x = 7 x 14 1212

88. Similar Figures Word Problems

89. Brayden reduced the size of a rectangle to a height of 2 inches. What is the new width if it was originally 24 inches wide and 12 inches tall?

90. Answer: ― = ― x = 4 inches 24 12 x2x2

91. You can use indirect measurement to measure distances that are difficult to measure directly by using proportions and similar figures.

92. A 6 ft tall person standing near a flagpole cast a shadow 4.5 ft long. The flagpole casts a shadow 15 ft long. What is the height of the flagpole?

94. Answer: ― = ― 4.5 ∙ x = 6 ∙ 15 x = (6 ∙ 15)/4.5 x = 20 x6x6 15 4.5

95. Scale Drawings

96. A scale drawing is an enlarged or reduced drawing of an object that is similar to the actual object. A scale is the ratio that compares a length in a drawing or model to the corresponding length in the actual object.

97. The distance of the Rio Grande is 3.5 inches on a map. If the map scale is ½ inch = 2 miles, what is the distance of the Rio Grande? STOP

98. Answer: ― = ― ½ ∙ (x) = 2 ∙ (3.5) ½ x = 7 x = 14 miles ½2½2 3.5 x

99. Competition Problems

100. Write 4 inches to 2 yards as a ratio in simplest form. STOP

101. Answer: 1/18

102. Solve the proportion:

103. Answer: 3

104. The scale of a map is 1 in. = 20 mi. Find the actual distance, in miles, represented by 2 ¾ inches on the map?

105. Answer: 55

106. Ford makes 7 red cars for every 4 white cars made. If it makes 16 white cars a day, how many red cars will it make a day?

107. Answer: 28

108. A telephone pole is 30 feet tall. It casts a shadow that is 24 feet long. A tree that is next to the telephone pole is 5 feet tall. How long is the shadow of the tree?

109. Answer: 4

110. If, solve for n.

111. Answer: 4 ⅖

112. A college football stadium holds 45,000 fans. In a random sample of 30 fans, 26 were wearing the colors of the home team. Predict the total number of fans who are wearing the colors of the home team.

113. Answer: 39,000

114. The distance between Earth and Planet Xeron is 5.274 × 10 8 miles. The Xerons make the trip in 3000 days. On average, how many miles per day do they travel?

115. Answer: 175,800

116. In a coastal region, there are 104 marked dolphins. In a sampling, biologists count 45 dolphins, of which 33 are marked. About how many dolphins are in the region?

117. Answer: 142

118. The ratio of girls to boys in a sixth grade art class is 6:5. If there are 121 students in the class, how many are boys?

119. Answer: 55

120. A recipe uses 1 ¾ cups of chocolate chips for every 24 cookies. How many cups of chocolate chips will be needed to make five dozen cookies?

121. Answer: 4 3 / 8

122. Triangle EFG is similar to triangle XYZ. Find the area of triangle XYZ.

123. Answer: 80

124. John is building a bird house. The scale is ¼ ft. to 1 ft. One wall is 2 ½ inches tall as drawn on the blueprint. What is the actual wall height?

125. Answer: 10 inches

126. The distance from Birmingham to Atlanta is 2.25 inches on a map. If the map scale is 1/4 inch=2 miles, what is the distance in miles?

127. Answer: 18 miles

128. Rebecca takes 20 minutes to cut a stick of bamboo into 5 pieces. How long does it take her to cut another stick into 9 pieces?

129. Answer: 40 minutes

130. If a quart is ¼ of a gallon and a pint is 1/8 of a gallon, how much of a quart is a pint?

131. Answer: 1/2

132. The clock in the bell tower rings every half hour. How many times will it ring in one week?

133. Answer: 336

134. If a worker makes 45 boxes of labels in 2 ½ hours, then at the same rate, how many boxes of labels can be made in 50 minutes?

135. Answer: 15

136. If 15 blips are equal to 7 bloops, 5 blumps are equal to 7 blatts, 4 bloops are equal to 10 blatts, then how many blips are equal to 30 blumps?

137. Answer: 36

138. Which of these represents the greatest amount of time? A.3 years B. 35 months C. 158 weeks D. 1096 days E. 26,200 hours

139. Answer: 158 weeks