This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Define variables and set up equations for the two parts of the journey. Let D be the total distance of the journey in km. For the first part of the journey: The distance covered is D_1 = (2)/(5)D. The speed is S_1 = 30 km/h. The time taken for the first part is T_1 = (D_1)/(S_1) = (2)/(5)D30 = (2D)/(150) = (D)/(75) hours. For the second part of the journey (remaining distance): The remaining distance is D_2 = D - D_1 = D - (2)/(5)D = (3)/(5)D. The speed is S_2 = 40 km/h. The time taken for the second part is T_2 = (D_2)/(S_2) = (3)/(5)D40 = (3D)/(200) hours. Step 2: Use the total time to form an equation. The total time for the whole journey is given as 4 hours. So, T_1 + T_2 = 4. (D)/(75) + (3D)/(200) = 4 Step 3: Solve the equation for D. To solve for D, find the least common multiple (LCM) of the denominators 75 and 200. 75 = 3 × 5^2 200 = 2^3 × 5^2 The LCM(75, 200) = 2^3 × 3 × 5^2 = 8 × 3 × 25 = 600. Multiply the entire equation by 600 to eliminate the denominators: 600 ( (D)/(75) ) + 600 ( (3D)/(200) ) = 4 × 600 8D + 3(3D) = 2400 8D + 9D = 2400 17D = 2400 D = (2400)/(17) km Step 4: Convert the fraction to a repeating decimal. Perform the division 2400 ÷ 17: 2400 ÷ 17 = 141 with a remainder of 3. So, D = 141 + (3)/(17). Now, convert the fraction (3)/(17) to a repeating decimal: (3)/(17) = 0.1764705882352941 The repeating block consists of 16 digits. Therefore, the total distance D is: D = 141.1764705882352941 km The total distance of the journey is 141.1764705882352941 km. Send me the next one 📸