Questions and Answers #22 Ratios, Rates, Proportions

Questions and Answers Sheet 22 Ratios, Rates, Proportions Question #1 The solution of the given proportion. Given: 𝑤 4.5 = 2 6.8 Answer: Calculation: The given ratios can be simplified further by doing cross product: 𝑤 2 = 4.5 6.8 w(6.8)=2(4.5) 6.8w=9 6.8𝑤 9 = 6.8 6.8 w=1.32 Question #2 To calculate: the total of stamps in the collection and the number of stamps from U.S. in the collection. Given information: 3 There are 15 stamps from Canada and they are of the total collection. 11 Answer: Calculation: Let, x be the total number of stamps. The number of stamps from Canada are 15. The number of stamps from U.S. are (x-15) . 3 Since, the Canada stickers made up of the total collection. 11 The total number of stamps in the collection can be calculate as, 15 𝑥 3 = 11 3 11 3𝑥 11 1 × 𝑥 = 15 [cross multiply] × 11 = 15 × 11 [multiply each side by 11] 3x=55 3𝑥 ÷ 3 = 165 ÷ 3 [divide each side by 3] x=55 The number of stamps from U.S. in the collection can be calculated as, (x-15)=(55-15) (x-15)=40 Therefore, The total number of stamps in the collection are 55 and the number of stamps from U.S. in the collection are 40. Question #3 To calculate: The equivalent rate of the given ratio. 5 𝑐𝑚 Given: 1 𝑚𝑖𝑛 Convert cm per min to cm per hour. Answer: Calculation: As per the given problem, Consider the rate: 5𝑐𝑚 1𝑚𝑖𝑛 To convert minutes into hour, multiply the rate by a conversion factor. 60𝑚𝑖𝑛 In 1 hour, there are 60 minutes, so 5𝑐𝑚 = 5𝑐𝑚 ⋅ 1𝑚𝑖𝑛 1𝑚𝑖𝑛 300𝑐𝑚 = 60𝑚𝑖𝑛 1ℎ𝑜𝑢𝑟 = 1. [Multiply rate by conversion factor.] 1ℎ𝑜𝑢𝑟 [Cancel out common units and simplify] 1ℎ𝑜𝑢𝑟 Therefore, 5𝑐𝑚 1𝑚𝑖𝑛 = 300𝑐𝑚 1ℎ𝑜𝑢𝑟 Hence, The equivalent rate of 5𝑐𝑚 1𝑚𝑖𝑛 is 300𝑐𝑚 1ℎ𝑜𝑢𝑟 . Question #4 To calculate: The equivalent rate of the given ratio. Given information: Consider the rate given in the equation: 3𝑙𝑏 1 Answer: Calculation: As per the given problem, Consider the rate: 3𝑙𝑏 1 To convert pounds into ounces, multiply the rate by a conversion factor. 16𝑜𝑧 1 pound (lb) is equal to 16 Ounces(oz), so =1 1𝑙𝑏 3𝑙𝑏 1 = = 3𝑙𝑏 48𝑜𝑧 1 1 ⋅ 16𝑜𝑧 [Multiply rate by conversion factor.] 1𝑙𝑏 [Divide out common unit and simplify.] Therefore, 1𝑙𝑏 1 = 48𝑜𝑧 1 Hence, The equivalent rate of 3𝑙𝑏 1 is 48𝑜𝑧 1 . Question #5 To determine whether the ratio is in simplest form and write the ratio in two other ways. Given: The ratio is 15 3 . Answer: Calculation: Simplify the ratio, 15 [highest common factor of 15 and 3 is 3] 3 = 15÷3 3÷3 [divide the numerator and denominator by 3] =5 Since, the simplest form of the ratio 15 3 Hence, the ratio is not in simplest form. is 5 . The ratio 15 3 15 3 = = 15×2 3×2 15×3 3×3 15 can also be written as 3 = = 30 [multiply the numerator and denominator by 2] 6 45 [multiply the numerator and denominator by 3] 9 Hence, the ratio 15 3 can also be written as 30 6 and 45 9 . Question #6 To determine whether the ratio is in simplest form and write the ratio in two other ways. Given: The ratio is 24 : 8 . Answer: Calculation: Simplify the ratio, 24 [highest common factor of 24 and 8 is 8] 8 24÷8 = 8÷8 3 = [divide the numerator and denominator by 8] 1 =3 : 1 Since, the simplest form of the ratio 24:8 is 3:1 . Hence, the ratio 24:8 is not in simplest form. 24 24×2 48 The ratio 24:8 can also be written as = = = 48: 16 [multiply the numerator and denominator by 8 8×2 16 2] 24 8 = 24×3 8×3 = 72 24 = 72: 24 [multiply the numerator and denominator by 3] Hence, the ratio 24:8 can also be written as 48:16 and 72:24. Question #7 To determine whether the ratio is in simplest form and write the ratio in two other ways. Given information: The ratio is 64:3 Answer: Calculation: Since, in the ratio 64:3 , 64 and 3 do not have any common factor. Hence, the ratio 64:3 is in simplest form. The ratio 64:3 can also be written as 64 64×2 128 = = = 128: 6 [multiply the numerator and denominator by 2] 3 64 3 = 3×2 64×3 3×3 = 6 192 9 = 192: 9 [multiply the numerator and denominator by 3] Hence, the ratio 64:3 can also be written as 128:6 and 192:9. Question #8 To determine the unit rate Given: The ratio is 161 7 𝑠ℎ𝑎𝑟𝑒𝑠 Answer: Calculation: The unit rate can be obtained by 161 7𝑠ℎ𝑎𝑟𝑒𝑠 = = 161÷7 7 𝑠ℎ𝑎𝑟𝑒𝑠÷7 23 [divide the numerator and denominator by 7] 1 𝑠ℎ𝑎𝑟𝑒 Hence, the unit rate is $23 per share. Question #9 To calculate: The equivalent rate. Given information: 300𝑚𝑖 20𝑠𝑒𝑐 = ?𝑚𝑖 1𝑚𝑖𝑛 Answer: Calculation: To find an equivalent rate, convert from miles per second to miles per minute. To do so, multiply the rate 60𝑠𝑒𝑐 by a conversion factor. There are 60 seconds in 1 minute, so = 1. 1𝑚𝑖𝑛 300𝑚𝑖 20𝑠𝑒𝑐 300𝑚𝑖 60𝑠𝑒𝑐 ⇒ × [Multiply rate by common conversion factor] 20𝑠𝑒𝑐 1𝑚𝑖𝑛 300mi 60sec   [Divide out common factor and unit] 20sec 1min ⇒ 900𝑚𝑖 1𝑚𝑖𝑛 [Simplify] Therefore, the equivalent rate is 900 miles per minute. Question #10 The radius r of a sphere is increasing at a rate of 6 inches per minute. (a) Find the rate of change of the volume when r=11 inches. (b) Find the rate of change of the volume when r=34 . Answer: Step 1 (a) Given: 𝑑𝑟 𝑑𝑡 = 6 inches r=11 inches Step 2 Find the rate of change of volume: 4 Volume of the sphere = 𝜋𝑟 3 3 𝑑𝑉 𝑑𝑡 4 = (3)𝜋𝑟 2 3 = 4𝜋𝑟 2 𝑑𝑟 𝑑𝑡 𝑑𝑟 𝑑𝑡 Substitute r=11 inches and 𝑑𝑉 𝑑𝑡 = 4𝜋(11)2 (6) = 2904𝜋 𝑖𝑛.3 /𝑚𝑖𝑛 𝑑𝑟 𝑑𝑡 = 6 inches in above equation