Two trains started simultaneously from opposite ends of a 100-mile route and traveled toward each other on parallel tracks. Train...

1. Translate the problem requirements: Two trains start simultaneously from opposite ends of a 100-mile route and travel toward each other. We need to find how far Train X has traveled when they meet, given their individual completion times.
2. Determine individual train speeds: Calculate each train's rate using the given completion times for the full 100-mile journey.
3. Set up the meeting condition: Since they start simultaneously and travel toward each other, they will meet when the sum of distances they've each traveled equals 100 miles.
4. Apply the time relationship: Both trains travel for the same amount of time until they meet, so use this common time to find Train X's distance.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what's happening in plain English. We have two trains starting at the same time from opposite ends of a 100-mile track. Think of it like two people walking toward each other from opposite ends of a hallway - they'll eventually meet somewhere in the middle.

Train X starts from one end and would take 5 hours to travel the full 100 miles if it went all the way. Train Y starts from the other end and would take 3 hours to travel the full 100 miles if it went all the way.

But here's the key: they don't travel the full distance because they meet each other partway through their journeys. We need to find exactly how far Train X has traveled when this meeting happens.

Process Skill: TRANSLATE - Converting the problem scenario into a clear mathematical situation

2. Determine individual train speeds

Now let's figure out how fast each train travels. Speed is simply distance divided by time.

Train X completes 100 miles in 5 hours, so its speed is:
Train X speed = \(100 \text{ miles} ÷ 5 \text{ hours} = 20 \text{ miles per hour}\)

Train Y completes 100 miles in 3 hours, so its speed is:
Train Y speed = \(100 \text{ miles} ÷ 3 \text{ hours} = 33\frac{1}{3} \text{ miles per hour}\) (or \(\frac{100}{3} \text{ miles per hour}\))

Notice that Train Y is faster than Train X, which makes sense since it can complete the same journey in less time.

3. Set up the meeting condition

Here's the crucial insight: when the trains meet, the total distance they've traveled together equals the original 100-mile separation.

Think about it this way: if Train X travels 30 miles from its starting point and Train Y travels 70 miles from its starting point, then together they've covered \(30 + 70 = 100 \text{ miles}\), which means they've met.

So our condition is: Distance traveled by Train X + Distance traveled by Train Y = 100 miles

4. Apply the time relationship

Both trains start at the same time and travel until they meet, so they both travel for exactly the same amount of time. Let's call this meeting time 't' hours.

Distance traveled by Train X = \(20\mathrm{t} \text{ miles}\) (speed × time)
Distance traveled by Train Y = \(\frac{100}{3}\mathrm{t} \text{ miles}\)

Using our meeting condition:
\(20\mathrm{t} + \frac{100}{3}\mathrm{t} = 100\)

To solve this, let's get a common denominator:
\(\frac{60}{3}\mathrm{t} + \frac{100}{3}\mathrm{t} = 100\)
\(\frac{160}{3}\mathrm{t} = 100\)
\(\mathrm{t} = 100 × \frac{3}{160} = \frac{300}{160} = \frac{15}{8} = 1.875 \text{ hours}\)

Therefore, Train X travels for 1.875 hours at 20 mph:
Distance traveled by Train X = \(20 × 1.875 = 37.5 \text{ miles}\)

Process Skill: MANIPULATE - Working with fractions and equations to find the solution

4. Final Answer

Train X had traveled 37.5 miles when it met Train Y.

Let's verify: Train Y traveled \(\frac{100}{3} × 1.875 = 62.5 \text{ miles}\)
Total: \(37.5 + 62.5 = 100 \text{ miles}\) ✓

The answer is A. 37.5

Common Faltering Points

Errors while devising the approach

Students often think each train travels the full 100 miles before meeting, rather than understanding that they start simultaneously from opposite ends and meet somewhere in between. This leads to incorrect setups where they try to find when one train "catches up" to another.

Students may mistakenly think the trains meet after 5 hours (Train X's completion time) or 3 hours (Train Y's completion time), instead of recognizing that these are the times it would take each train to complete the full journey alone. The actual meeting time is different and needs to be calculated.

Some students might set up the equation as "Distance of X - Distance of Y = 100" or use other incorrect relationships, rather than understanding that the sum of distances traveled by both trains equals the total separation distance of 100 miles.

Errors while executing the approach

When working with Train Y's speed of \(\frac{100}{3} \text{ mph}\) and solving the equation \(20\mathrm{t} + \frac{100}{3}\mathrm{t} = 100\), students often make mistakes in finding common denominators or simplifying fractions. They might incorrectly calculate \(\frac{60}{3}\mathrm{t} + \frac{100}{3}\mathrm{t}\) as something other than \(\frac{160}{3}\mathrm{t}\).

When solving for \(\mathrm{t} = 100 × \frac{3}{160}\), students may make computational errors, getting values other than 1.875 hours. Common mistakes include getting 1.5 hours or 2.5 hours due to incorrect fraction operations.

Even if they find the correct meeting time (1.875 hours), students might accidentally use Train Y's speed instead of Train X's speed, calculating \(\frac{100}{3} × 1.875 = 62.5 \text{ miles}\) instead of \(20 × 1.875 = 37.5 \text{ miles}\).

Errors while selecting the answer

Students might correctly calculate that Train X travels 37.5 miles and Train Y travels 62.5 miles, but then select 62.5 miles (choice D) instead of 37.5 miles (choice A) because they lose track of which train the question is asking about.