A car travels 120 km at an average speed of 60 km/h, then travels another 180 km at 90 km/h. What is the weighted average speed for the entire journey in km/h?
["Weighted Average Speed: How to Calculate and Apply It in a Two-Part Journey", "When analyzing multi-stage journeys, one key question often arises: What is the true average speed for the entire trip? In this article, we explore a classic example—計算平均速度 — where a car travels 120 km at 60 km/h, then another 180 km at 90 km/h. Understanding how to compute the weighted average speed reveals not just the math, but the importance of time in measuring travel efficiency.", "---", "### The Journey Overview", "The car’s journey consists of two distinct segments:\n- Segment 1: 120 km at 60 km/h\n- Segment 2: 180 km at 90 km/h", "Rather than simply averaging speeds using (\frac{\ ext{Total Distance}}{\ ext{Total Time}}) directly (though tempting), the correct approach uses the weighted average speed based on time spent in each segment.", "---", "### Step 1: Compute Time for Each Segment", "Speed = Distance ÷ Time → Time = Distance ÷ Speed", "- First segment (120 km at 60 km/h):\n [\n \ ext{Time}_1 = \frac{120\ \ ext{km}}{60\ \ ext{km/h}} = 2\ \ ext{hours}\n ]", "- Second segment (180 km at 90 km/h):\n [\n \ ext{Time}_2 = \frac{180\ \ ext{km}}{90\ \ ext{km/h}} = 2\ \ ext{hours}\n ]", "---", "### Step 2: Total Distance and Total Time", "- Total Distance:\n [\n 120\ \ ext{km} + 180\ \ ext{km} = 300\ \ ext{km}\n ]", "- Total Time:\n [\n 2\ \ ext{hours} + 2\ \ ext{hours} = 4\ \ ext{hours}\n ]", "---", "### Step 3: Calculate Weighted Average Speed", "[\n\ ext{Weighted Average Speed} = \frac{\ ext{Total Distance}}{\ ext{Total Time}} = \frac{300\ \ ext{km}}{4\ \ ext{hours}} = 75\ \ ext{km/h}\n]", "Alternatively, using the weighted average formula:", "[\n\ ext{Weighted Speed} = \frac{(120\ \ ext{km} \ imes 60\ \ ext{km/h}) + (180\ \ ext{km} \ imes 90\ \ ext{km/h})}{120\ \ ext{km} + 180\ \ ext{km}} = \frac{7200 + 16200}{300} = \frac{23400}{300} = 75\ \ ext{km/h}\n]", "Both methods confirm the same result: 75 km/h.", "---", "### Why Weighted Average Matters", "Averaging speeds using only distances (e.g., (\frac{120 + 180}{2} = 150) km/h) ignores the fact that time is the true driving factor of average speed. Faster travel times reduce the average, so weighted calculation reveals the actual effective pace.", "In real-world applications—whether for planning road trips, logistics routes, or analyzing vehicle performance—weighted average speed offers the most accurate insight.", "---", "### Conclusion", "For a 120 km segment at 60 km/h followed by an 180 km segment at 90 km/h, the weighted average speed across the full journey is 75 km/h—a clear example of how time-weighted averages deliver meaningful transportation insights.", "---", "Key Takeaways:\n- Use time in each segment to calculate a proper weighted average speed.\n- The formula:\n [\n \ ext{Weighted Average Speed} = \frac{\sum (\ ext{Distance}_i \ imes \ ext{Speed}_i)}{\sum \ ext{Distance}_i}\n ]\n- Average speed based on distance alone is misleading—time is essential.\n- This concept helps optimize travel planning and performance evaluation for vehicles.", "Keywords: average speed, weighted average speed, car journey calculation, travel speed analysis, physics problem solution, road trip planning", "---", "Understanding how to compute and interpret the weighted average speed transforms routine trip planning into precise, data-driven decisions. Whether driving cross-country or managing a delivery fleet, mastering this concept ensures smarter, faster, and more efficient journeys."]
