Questions and Answers #6 Ratios, Rates, Proportions

Questions and Answers Sheet 6 Ratios, Rates, Proportions Question #1 To calculate: The number of cups of water required for water for every 1 2 3 4 cup of oats if cooking oatmeal requires 1 cup of cup of oats. Answer: Given information: The Cooking oatmeal requires 1 cup of water for every 1 2 cup of oats. Formula used: An equation that equates two ratios or rates is called a proportion. Therefore, if 𝑏 ≠ 0 and 𝑑 ≠ 0 , then: 𝑎 𝑐 = is a proportion. 𝑏 𝑑 Calculation: As it is provided that cooking oatmeal requires 1 cup of water for every Consider x represent the number of cups of water required for 3 4 1 2 cup of oats. cup of oats. Set up a proportion by writing two equivalent ratios: 1 𝑥 1 = 3 2 4 3 Multiply by LCD, that is : 3 8 3 4 ⋅ 1 1 2 = = 𝑥 𝑥 3 4 ⋅ 8 3 8 2 6=4x Solve the resulting linear equation as, 4x=6 3 𝑥= 2 Hence, the necessary required water is 1.5 cups. Question #2 Whether the proportion 7 4 = 5 3 is true. Answer: Approach: The quotient of two numbers is called a ratio. 2 Ex. . 3 The ratio is no different from a fraction, except that a ratio is sometimes written using notation other than fractional notation. 5 Ex. The ratio of 5 to 7 can be written as fractional notation or colon notation 5:7. 7 If a ratio contains decimal numbers or mixed numbers, then the ratio can be written only by the ratio of whole numbers. 57 Ex. 5.7 : 3 can be written as 30 A proportion is a statement that two ratios or rates are equal. 𝑎 𝑐 𝑎 𝑐 Ex. If and are two ratios, then = is a proportion. 𝑏 𝑑 𝑏 𝑑 If the cross products are not equal, the proportion is true. Ex, If 𝑎 ⋅ 𝑑 = 𝑏 ⋅ 𝑐 , then it is true. If the cross products are not equal, the proportion is false. Ex. If 𝑎 ⋅ 𝑑 ≠ 𝑏 ⋅ 𝑐 , then it is false. Calculation: 7 5 Determine the proportion = is true or false by using the cross product. 4 7 4 = 3 5 3 7 ⋅4=5 ⋅3 28 ≠ 15 Therefore, the cross product is not equal. Hence, the statement is false. The final Statement: The proportion 7 4 = 5 3 is not true. Question #3 Whether the proportion 8.2 2 = 16.4 4 is true. Answer: Approach: The quotient of two numbers is called a ratio. 2 Ex. . 3 The ratio is no different from a fraction, except that a ratio is sometimes written using notation other than fractional notation. 5 Ex. The ratio of 5 to 7 can be written as fractional notation or colon notation 5:7. 7 If a ratio contains decimal numbers or mixed numbers, then the ratio can be written only by the ratio of whole numbers. Ex. 5.7 : 3 can be written as 57 30 A proportion is a statement that two ratios or rates are equal. 𝑎 𝑐 𝑎 𝑐 Ex. If and are two ratios, then = is a proportion. 𝑏 𝑑 𝑏 𝑑 If the cross products are not equal, the proportion is true. Ex, If 𝑎 ⋅ 𝑑 = 𝑏 ⋅ 𝑐, then it is true. If the cross products are not equal, the proportion is false. Ex. If 𝑎 ⋅ 𝑑 ≠ 𝑏 ⋅ 𝑐 , then it is false. Calculation: 8.2 16.4 Determine the proportion = is true or false by using the cross product. 8.2 2 = 2 16.4 4 4 8.2 ⋅ (4) = 16.4 ⋅ (2) 32.8=32.8 Therefore, the cross product is equal. Hence, the statement is true. The final Statement: 8.2 16.4 The proportion = is true. 2 4 Question #4 To solve: The proportion 5 3 = 40 𝑥 . Answer: Approach: The quotient of two numbers is called a ratio. 2 Ex. . 3 The ratio is no different from a fraction, except that a ratio is sometimes written using notation other than fractional notation. Ex. The ratio of 5 to 7 can be written as fractional notation 5 7 or colon notation 5:7 . If a ratio contains decimal numbers or mixed numbers, then the ratio can be written only by the ratio of whole numbers. 57 Ex. 5.7 : 3 can be written as 30 A proportion is a statement that two ratios or rates are equal. 𝑎 𝑐 𝑎 𝑐 Ex. If and are two ratios, then = is a proportion. 𝑏 𝑑 𝑏 𝑑 If the cross products are not equal, the proportion is true. Ex, If 𝑎 ⋅ 𝑑 = 𝑏 ⋅ 𝑐 , then it is true. If the cross products are not equal, the proportion is false. Ex. If 𝑎 ⋅ 𝑑 ≠ 𝑏 ⋅ 𝑐 , then it is false. When one number of proportion is unknown, then use cross products to find the unknown number. 4 20 Ex. Find the unknown number x in the proportion = . It can be found by cross product. That is, x=10 . 2 Calculation: Determine the value of x in the proportion 5 3 = 5 3 = 40 𝑥 𝑥 by using the cross product. 40 𝑥 5 ⋅ 𝑥 = 40 ⋅ 3 120 𝑥= 5 x=24 Therefore, the value is x=24 . The final Statement: The value of x in the proportion 5 3 = 40 𝑥 is x=24. Question #5 To solve proportion 21 𝑥 7 = 2. 3 Answer: Approach: The quotient of two numbers is called a ratio. 2 Ex . 3 The ratio is no different from a fraction, except that a ratio is sometimes written using notation other than fractional notation. 5 Ex. The ratio of 5 to 7 can be written as fractional notation or colon notation 5:7 . 7 If a ratio contains decimal numbers or mixed numbers, then the ratio can be written only by the ratio of whole numbers. Ex. 5.7 : 3 can be written as 57 30 A proportion is a statement that two ratios or rates are equal. 𝑎 𝑐 𝑎 𝑐 Ex. If and are two ratios, then = is a proportion. 𝑏 𝑑 𝑏 𝑑 If the cross products are not equal, the proportion is true. Ex, If 𝑎 ⋅ 𝑑 = 𝑏 ⋅ 𝑐 , then it is true. If the cross products are not equal, the proportion is false. Ex. If 𝑎 ⋅ 𝑑 ≠ 𝑏 ⋅ 𝑐 , then it is false. When one number of proportion is unknown, then use cross products to find the unknown number. 4 20 Ex. Find the unknown number x in the proportion = . It can be found by cross product. That is, x=10 . 2 𝑥 Calculation: Determine the value of x in the proportion 21 𝑥 = 7 2 3 7 21 ⋅ 3 = ⋅ 𝑥 2 21 𝑥 = 7 2 3 by using the cross product. 7 63 = ⋅ 𝑥 2 2 On further simplification, 𝑥 = 63 ⋅ 7 126 𝑥= 7 x=18 Therefore, the value is x=18 . The final Statement: 21 The value of x in the proportion 𝑥 = 7 2 3 is x=18 . Question #6 To solve: The proportion 𝑦 10 = 13 4 Answer: Approach: The quotient of two numbers is called a ratio. 2 Ex. . 3 The ratio is no different from a fraction, except that a ratio is sometimes written using notation other than fractional notation. 5 Ex. The ratio of 5 to 7 can be written as fractional notation or colon notation 5:7. 7 If a ratio contains decimal numbers or mixed numbers, then the ratio can be written only by the ratio of whole numbers. Ex. 5.7 : 3 can be written as 57 30 A proportion is a statement that two ratios or rates are equal. 𝑎 𝑐 𝑎 𝑐 Ex. If and are two ratios, then = is a proportion. 𝑏 𝑑 𝑏 𝑑 If the cross products are not equal, the proportion is true. Ex, If 𝑎 ⋅ 𝑑 = 𝑏 ⋅ 𝑐 , then it is true. If the cross products are not equal, the proportion is false. Ex. If 𝑎 ⋅ 𝑑 ≠ 𝑏 ⋅ 𝑐, then it is false. When one number of proportion is unknown, then use cross products to find the unknown number. 4 20 Ex. Find the unknown number x in the proportion = . It can be found by cross product. That is, x=10 . 2 Calculation: Determine the value of x in the proportion 𝑦 10 = 13 𝑦 10 = 13 4 𝑥 by using the cross product. 4 𝑦 ⋅ 4 = 13 ⋅ 10 130 = 4 y=32.5 Therefore. the value is y=32.5 The final Statement: The value of y in the proportion 𝑦 10 = 13 4 is y=32.5. Question #7 To solve: the given proportion: 9 7 = 𝑏+11 14 Answer: Calculation: A proportion is an equation stating that the two ratios or rates are equal. Simplify 9 7 = 𝑏+11 14 as follows: (𝑏 + 11) × 7 = 9 × 14 𝑏 × 7 + 11 × 7 = 126 7𝑏 + 77 = 126 7𝑏 + 77 − 77 = 126 − 77 [Add 77 on both sides] 7b=49 7𝑏 7 = 49 7 [Divide each side by 7] b=7 Hence, the value of b is b=7 . Question #8 The given proportion: 𝑎 0.28 = 4 1.4 Answer: Calculation: A proportion is an equation stating that the two ratios or rates are equal. 𝑎 4 Simplify = as follows: 0.28 1.4 𝑎(1.4) = 4(0.28) [The cross products of any proportion are equal] 1.4𝑎 = 1.12 [Multiply] 1.4𝑎 1.12 = [Divide each side by 1.4] 1.4 1.4 a=0.8 Hence, the value of a is 0.8. Question #9 The given proportion: 𝑥−18 24 = 15 8 Answer: Calculation: A proportion is an equation stating that the two ratios or rates are equal. 𝑥−18 15 Simplify = as follows: 24 8 (𝑥 − 18) × 8 = 15 × 24 (𝑥 − 18) − (18 × 8) = 360 8x-144=360 8x-144+144=360+144 [Add 128 on both sides] 8x=504 8𝑥 8 = 504 8 [Divide each side by 8] x=63 Hence, the value of x is x=63 . Question #10 The given proportion: 16 𝑥+5 = 4 5 Answer: Calculation: A proportion is an equation stating that the two ratios or rates are equal. 16 4 Simplify = as follows: 𝑥+5 5 16 × 5 = 4 × (𝑥 + 5) 80 = 4 × 𝑥 + 4 × 5 80=4x+20 80 − 20 = 4𝑥 + 20 − 20 [Add (-20) on both sides] 60=4x 60 4𝑥 = [Divide each side by 4] 4 4 15=x Hence, the value of x is x=15