A train travels 300 km at a speed of 60 km/h and then another 200 km at a speed of 80 km/h. What is the train's average speed for the entire journey?
["Title: How to Calculate Average Speed for a Train Journey: A Practical Example", "When traveling by train, one common question travelers and commuters ask is: What is the average speed for the entire trip? Understanding this concept helps in planning travel time and making informed decisions. Let’s explore a practical example: when a train travels 300 kilometers at 60 km/h, followed by another 200 kilometers at 80 km/h, what is the overall average speed?", "---", "### Understanding Average Speed", "Average speed is not simply the mean of the two speeds. Instead, it’s the total distance traveled divided by the total time taken. This accounts for the varying speeds over different segments of the journey.", "Mathematically:", "[\n\ ext{Average Speed} = \frac{\ ext{Total Distance}}{\ ext{Total Time}}\n]", "---", "### Step 1: Calculate Total Distance", "The train travels two legs:", "- First segment: 300 km\n- Second segment: 200 km", "[\n\ ext{Total Distance} = 300,\ ext{km} + 200,\ ext{km} = 500,\ ext{km}\n]", "---", "### Step 2: Calculate Travel Time for Each Segment", "Time = Distance ÷ Speed", "- For the first 300 km at 60 km/h:\n[\n\ ext{Time}_1 = \frac{300}{60} = 5,\ ext{hours}\n]", "- For the next 200 km at 80 km/h:\n[\n\ ext{Time}_2 = \frac{200}{80} = 2.5,\ ext{hours}\n]", "---", "### Step 3: Find Total Time", "[\n\ ext{Total Time} = 5,\ ext{hours} + 2.5,\ ext{hours} = 7.5,\ ext{hours}\n]", "---", "### Step 4: Compute Average Speed", "[\n\ ext{Average Speed} = \frac{500,\ ext{km}}{7.5,\ ext{hours}} = 66.\overline{6}, \ ext{km/h} \approx 66.67,\ ext{km/h}\n]", "---", "### Why This Method Matters", "Relying on a simple arithmetic mean—like (60 + 80)/2 = 70 km/h—would misrepresent the actual average because the train spends more time at the slower speed. This example highlights that time spent at each speed plays a critical role in determining average speed.", "---", "### Conclusion", "To find the average speed for a multi-segment journey, calculate the total distance and total time, then divide:", "[\n\ ext{Average Speed} = \frac{\ ext{Total Distance}}{\ ext{Total Time}} = \frac{500}{7.5} \approx 66.67,\ ext{km/h}\n]", "Next time your train travels different distances at different speeds, remember: average speed reveals how long your trip truly took, not just what speeds you hit along the way.", "---", "Keywords: average speed, train journey, calculate average speed, train travel time, average speed formula, transport efficiency, travel planning, speed vs time, transportation math"]
