A train travels from City A to City B at 60 km/h and returns at 90 km/h. If the total travel time is 5 hours, find the distance between the two cities.

["Title: How to Calculate the Distance Between Two Cities Using Varying Train Speeds – A 5-Hour Round Trip Journey", "When planning a train journey between two cities, variations in travel speeds on the outbound and return trips often puzzle travelers and problem solvers alike. One classic example involves a train traveling from City A to City B at 60 km/h and returning at 90 km/h, with a total round-trip time of 5 hours. But how do you find the actual distance between these cities? This article breaks down the math and logic behind solving this common travel optimization problem.", "### The Scenario", "- Speed from City A to City B: 60 km/h\n- Speed from City B to City A: 90 km/h\n- Total travel time: 5 hours\n- Unknown: Distance between City A and City B (let’s call it D)", "This problem reflects real-world dynamics where speed changes affect travel time—relevant not just for trains, but for commuters, logistics planners, and anyone analyzing efficient transportation.", "### Understanding the Trip", "The train covers the same distance D both ways. The time taken for each leg of the journey depends on speed:", "- Time from A to B: $ \ ext{Time}_1 = \frac{D}{60} $ hours\n- Time from B to A: $ \ ext{Time}_2 = \frac{D}{90} $ hours", "The total time is the sum of these two times:", "$$\n\frac{D}{60} + \frac{D}{90} = 5\n$$", "### Step-by-Step Solution", "1. Find a common denominator for the fractions:", "LCM of 60 and 90 is 180.", "$$\n\frac{D}{60} = \frac{3D}{180}, \quad \frac{D}{90} = \frac{2D}{180}\n$$", "2. Add the fractions:", "$$\n\frac{3D}{180} + \frac{2D}{180} = \frac{5D}{180} = 5\n$$", "3. Set up the equation:", "$$\n\frac{5D}{180} = 5\n$$", "4. Solve for D:", "Multiply both sides by 180:", "$$\n5D = 900\n$$", "Then divide by 5:", "$$\nD = \frac{900}{5} = 180\n$$", "### Result", "The distance between City A and City B is 180 kilometers.", "### Why This Matters", "This calculation uses the fundamental relationship between distance = speed × time, applied across equal but differently paced trips. It demonstrates how speed inversion affects total travel duration and efficiently solves real-world itinerary planning—whether for train schedules, road trips, or freight logistics.", "### Real-World Tips", "- Know your average speed when journey times differ significantly.\n- Use the formula for total time: $ \frac{D}{v_1} + \frac{D}{v_2} = T $\n- It helps in training vehicles, optimizing public transit, and managing commute schedules.", "### Key Takeaway", "For a round trip where one leg moves at 60 km/h and the return at 90 km/h, with a total time of 5 hours, the distance between City A and City B is exactly 180 km.", "---", "Keywords: train travel distance calculation, total travel time problem, round trip speed difference, how to calculate distance train journey, speed vs time travel problem, 60 km/h to 90 km/h train distance, distance between two cities formula, time and speed train journey.", "---", "Whether you're commuting between twin cities or studying transport efficiency, understanding the interplay between speed, time, and distance turns complex journeys into clear, solvable math."]