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A train traveled a certain route that consisted of 3 segments, all having exactly the same length. If the average speeds of the train in these segments were \(60\mathrm{km/h}\), \(120\mathrm{km/h}\), \(60\mathrm{km/h}\), what was the average speed of the train, in \(\mathrm{km/h}\), over the entire route?
Let's start by understanding what we're looking for in plain English. We have a train that travels a route with 3 parts, and each part is exactly the same distance. The train goes at different speeds in each part: 60 km/h in the first part, 120 km/h in the second part, and 60 km/h in the third part.
We want to find the average speed for the entire journey. Average speed is simply: how much total distance did we cover divided by how much total time did it take?
This is different from just averaging the three speeds \((60 + 120 + 60) \div 3 = 80\), because the train spends different amounts of time in each segment even though the distances are equal.
Process Skill: TRANSLATE - Converting the problem language into a clear mathematical understanding
Since all three segments have the same length, let's pick a convenient distance that will make our math easy. Let's say each segment is 120 kilometers long.
Why 120 km? Because it divides evenly by both 60 and 120, which are our speeds. This will help us avoid messy fractions.
So our journey looks like this:
Now let's figure out how long each segment takes. We use the basic relationship: \(\mathrm{time} = \mathrm{distance} \div \mathrm{speed}\)
Segment 1: \(\mathrm{time} = 120 \text{ km} \div 60 \text{ km/h} = 2 \text{ hours}\)
Segment 2: \(\mathrm{time} = 120 \text{ km} \div 120 \text{ km/h} = 1 \text{ hour}\)
Segment 3: \(\mathrm{time} = 120 \text{ km} \div 60 \text{ km/h} = 2 \text{ hours}\)
Total time = \(2 + 1 + 2 = 5 \text{ hours}\)
Notice how the train spends more time in the slower segments (2 hours each) compared to the faster segment (1 hour). This is why we can't just average the speeds directly.
Now we can find the average speed for the complete journey:
Average speed = Total distance ÷ Total time
Average speed = \(360 \text{ km} \div 5 \text{ hours} = 72 \text{ km/h}\)
Let's double-check this makes sense: The train spends more time going slowly (4 hours at 60 km/h) than going fast (1 hour at 120 km/h), so the average should be closer to 60 km/h than to 120 km/h. Indeed, 72 km/h is much closer to 60 than to 120.
The average speed of the train over the entire route is 72 km/h.
Looking at our answer choices:
The answer is B. 72
1. Confusing average speed with arithmetic mean of speeds
Students often think that since the segments are equal in distance, they can simply calculate \((60 + 120 + 60) \div 3 = 80 \text{ km/h}\). This is incorrect because average speed requires total distance divided by total time, not just averaging the individual speeds.
2. Misunderstanding what 'equal segments' means
Some students might misinterpret 'equal segments' as equal time periods rather than equal distances. This would lead to a completely different calculation approach where they would incorrectly weight each speed equally by time.
3. Thinking the problem cannot be solved without knowing actual distances
Students may get stuck thinking they need the specific distance values to solve the problem, not realizing they can use any convenient distance (like 120 km) or work with variables since the segments are equal.
1. Arithmetic errors in time calculations
When calculating \(\mathrm{time} = \mathrm{distance} \div \mathrm{speed}\), students might make simple division errors. For example, calculating \(120 \div 60 = 3 \text{ hours}\) instead of 2 hours, or mixing up which numbers to divide.
2. Using inconsistent distance values
If students choose to use a variable approach or different distance values, they might inconsistently apply the same distance to all three segments, leading to incorrect time calculations.
3. Forgetting to add all time segments
Students might correctly calculate individual segment times but then forget to include one of the segments when calculating total time, leading to an incorrect final average speed.
1. Choosing the arithmetic mean (80) as the final answer
Even after doing the correct calculation and getting 72 km/h, some students might second-guess themselves and select choice C (80) because it matches their initial intuitive calculation of averaging the three speeds.
2. Selecting 'Cannot be determined' due to lack of confidence
Students who struggled with the concept might lose confidence in their answer and choose option E, thinking that without specific distance values, the problem cannot be solved.
Step 1: Choose a smart number for segment length
Since all three segments have equal length, let's choose 120 km for each segment. This number is strategically selected because it's divisible by all the given speeds (60 and 120), which will give us clean time calculations without fractions.
Step 2: Calculate time for each segment using \(\mathrm{time} = \mathrm{distance} \div \mathrm{speed}\)
Step 3: Calculate totals for the entire journey
Step 4: Apply average speed formula
Average speed = Total distance ÷ Total time = \(360 \text{ km} \div 5 \text{ hours} = 72 \text{ km/h}\)
Why this smart number works: By choosing 120 km (the LCM of the denominators when we express the speeds), we avoid fractional time calculations entirely, making the arithmetic straightforward while maintaining the same proportional relationships that determine the final answer.