Fractions #18 - Questions and Answers

Question 1 In the following exercises, approximate the (a) circumference and (b) area of each circle. If measurements are given in fractions, leave the answer in fraction form. Radius = 18.6m a) Approximate the circumference of the circle with radius 18.6m. b) Approximate the area of the circle with radius 18.6m. Answer 1 a) Formula used: Circumference of the circle 𝐶 = 2 𝜋 𝑟 C=2πr Where 𝑟 r is the radius of the circle. Calculation: We are given the radius of the circle as 18.6m. Substituting 𝜋 = 3.14 π=3.14 and 𝑟 = 18.6 r=18.6 into the circumference formula 𝐶 = 2 𝜋 𝑟 C=2πr, we get 𝐶 ≈ 2 ( 3.14 ) ( 18.6 ) C≈2(3.14)(18.6) On multiplying, we get 𝐶 ≈ 116.808 𝑚 . C≈116.808m. Conclusion: 𝐶 ≈ 116.808 𝑚 C≈116.808m b) Formula used: Area of the circle 𝐴 = 𝜋 𝑟 2 A=πr 2 Where 𝑟 r is the radius of the circle. Calculation: Substituting 𝜋 = 3.14 π=3.14 and 𝑟 = 18.6 r=18.6 into the formula for area 𝐴 = 𝜋 𝑟 2 A=πr 2 , we get 𝐴 ≈ ( 3.14 ) ( 18.6 ) 2 A≈(3.14)(18.6) 2 On simplifying the exponent ( 18.6 ) 2 (18.6) 2 , 𝐴 ≈ ( 3.14 ) ( 345.96 ) A≈(3.14)(345.96) On multiplying, we get 𝐴 ≈ 1086.3144 𝑚 2 A≈1086.3144m 2 Conclusion: 𝐴 ≈ 1086.3144 𝑚 2 A≈1086.3144m 2 Question 2 A line with a slope of 0 passes through the point (0,9). What is its equation in slope-intercept form? Write your answer using integers, proper fractions, and improper fractions in simplest form. Answer 1 Step 1: Given: To write the equation of a line in slope-intercept form that has slope 0 and passes through (0,9). Step 2: Let the equation of the required line be 𝑦 = 𝑚 𝑥 + 𝑐 y=mx+c, where 𝑚 m is the slope. The equation of the line becomes 𝑦 = 0 𝑥 + 𝑐 y=0x+c, which is 𝑦 = 𝑐 y=c. Since the line passes through (0,9), putting 𝑥 = 0 x=0 and 𝑦 = 9 y=9, we get 9 = 𝑐 9=c. Therefore, the equation of the line in slope-intercept form is: 𝑦 = 9 y=9 Question 3 Express each repeating decimal as a fraction. a) 5.23 b) 1.37 c) 2.135 Answer 1 Step 1: Definition used: Converting repeating decimal to fraction: First, we will equate the given decimal number to 𝑥 x and subtract the integer part from it, then multiply by 10, 100, or 1000 on both sides according to repeating digits. We solve for 𝑥 x and simplify fractions. Step 2: a) 5.23 Let 𝑥 = 5.23 x=5.23 𝑥 − 5 = 0.232323 … ( 1 ) x−5=0.232323…(1) Multiply both sides by 100: 100 ( 𝑥 − 5 ) = 23.232323 … ( 2 ) 100(x−5)=23.232323…(2) Subtracting equation (1) from equation (2): 99 ( 𝑥 − 5 ) = 23 99(x−5)=23 Solving for 𝑥 x: 𝑥 − 5 = 23 99 x−5= 99 23 ​ 𝑥 = 5 + 23 99 = 518 99 x=5+ 99 23 ​ = 99 518 ​ Step 3: b) 1.37 Let 𝑥 = 1.37 x=1.37 𝑥 − 1 = 0.373737 … ( 1 ) x−1=0.373737…(1) Multiply both sides by 100: 100 ( 𝑥 − 1 ) = 37.373737 … ( 2 ) 100(x−1)=37.373737…(2) Subtracting equation (1) from equation (2): 99 ( 𝑥 − 1 ) = 37 99(x−1)=37 Solving for 𝑥 x: 𝑥 = 1 + 37 99 = 136 99 x=1+ 99 37 ​ = 99 136 ​ Step 4: c) 2.135 Let 𝑥 = 2.135 x=2.135 𝑥 − 2.1 = 0.0353535 … ( 1 ) x−2.1=0.0353535…(1) Multiply by 1000: 1000 ( 𝑥 − 2.1 ) = 35.3535 … ( 2 ) 1000(x−2.1)=35.3535…(2) Subtracting equation (1) from equation (2): 900 ( 𝑥 − 2.1 ) = 35 900(x−2.1)=35 𝑥 = 2.1 + 35 990 = 495 211 x=2.1+ 990 35 ​ = 211 495 ​ Question 4 Which of the following uses the reciprocal property to rewrite 12 𝑥 − 3 = 20 𝑥 + 3 x−3 12 ​ = x+3 20 ​ ? Answer 1 Step 1: Given, The equation, 12 𝑥 − 3 = 20 𝑥 + 3 x−3 12 ​ = x+3 20 ​ Step 2: The reciprocal property is: If two fractions are equal, then their reciprocals are also equal. That is, 𝑎 𝑏 = 𝑐 𝑑 then 𝑏 𝑎 = 𝑑 𝑐 b a ​ = d c ​ then a b ​ = c d ​ Step 3: The reciprocal of 12 𝑥 − 3 x−3 12 ​ is 𝑥 − 3 12 12 x−3 ​ . The reciprocal of 20 𝑥 + 3 x+3 20 ​ is 𝑥 + 3 20 20 x+3 ​ . Therefore, by using the reciprocal property: 𝑥 − 3 12 = 𝑥 + 3 20 12 x−3 ​ = 20 x+3 ​ Thus, 𝑥 − 3 12 = 𝑥 + 3 20 12 x−3 ​ = 20 x+3 ​ uses the reciprocal property to rewrite 12 𝑥 − 3 = 20 𝑥 + 3 x−3 12 ​ = x+3 20 ​ . Question 5 Find the sum by converting the mixed numbers to improper fractions: 5 5 9 + 3 7 9 5 9 5 ​ +3 9 7 ​ Answer 1 Calculation: Given, 5 5 9 + 3 7 9 5 9 5 ​ +3 9 7 ​ Writing the mixed numbers as improper fractions: 5 5 9 = 50 9 , 3 7 9 = 32 9 5 9 5 ​ = 9 50 ​ ,3 9 7 ​ = 9 32 ​ To add or subtract rational numbers, the denominators need to be equal. When the denominators are equal, the numerators can be added or subtracted as per the operation. Hence, 50 9 + 32 9 = 82 9 9 50 ​ + 9 32 ​ = 9 82 ​ To convert the improper fraction to a mixed fraction, the numerator is written as the sum of a multiple of the denominator and the remainder. 82 9 = 9 1 9 9 82 ​ =9 9 1 ​ Therefore, 5 5 9 + 3 7 9 = 9 1 9 5 9 5 ​ +3 9 7 ​ =9 9 1 ​ .