The distance between Town R and Town S is 209 miles. At noon yesterday, a motorist started from Town R...

1. Translate the problem requirements: Two motorists start simultaneously from opposite towns (R and S) and drive toward each other at different constant speeds. We need to find how far from Town R they are when their paths cross.
2. Establish the meeting relationship: Since both motorists start at the same time and drive toward each other, they will meet when the sum of distances they've traveled equals the total distance between towns.
3. Set up time-based equations: Both motorists travel for the same amount of time until they meet, so we can use this common time to find when and where they meet.
4. Calculate the meeting point distance: Use the motorist from Town R's speed and the meeting time to determine how far from Town R the meeting occurs.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. We have two motorists who start driving toward each other at exactly the same time (noon yesterday). One starts from Town R and drives toward Town S at 40 mph. The other starts from Town S and drives toward Town R at 55 mph. The towns are 209 miles apart.

Think of it like two people walking toward each other from opposite ends of a long hallway - they'll meet somewhere in the middle, but not exactly in the middle because they're moving at different speeds.

We need to find: How far from Town R are the motorists when they meet?

Process Skill: TRANSLATE - Converting the real-world scenario into a clear mathematical setup

2. Establish the meeting relationship

Here's the key insight: When two objects start at the same time and move toward each other, they meet when the total distance they've both traveled equals the distance between their starting points.

Let's say the motorist from Town R travels a distance of 'd' miles before they meet. Then the motorist from Town S must have traveled (209 - d) miles before they meet, because together they must cover the full 209 miles.

This gives us our fundamental relationship:
Distance traveled by R motorist + Distance traveled by S motorist = 209 miles

Process Skill: VISUALIZE - Seeing how the two distances must add up to the total

3. Set up time-based equations

Since both motorists start at the same time and meet at the same time, they travel for exactly the same amount of time. Let's call this time 't' hours.

For the motorist starting from Town R:
- Speed = 40 mph
- Time = t hours
- Distance = 40t miles

For the motorist starting from Town S:
- Speed = 55 mph
- Time = t hours (same time!)
- Distance = 55t miles

Now we can use our relationship from step 2:
\(40\mathrm{t} + 55\mathrm{t} = 209\)
\(95\mathrm{t} = 209\)
\(\mathrm{t} = 209 ÷ 95 = 2.2\) hours

4. Calculate the meeting point distance

Now that we know they meet after 2.2 hours, we can find how far from Town R this meeting point is.

The motorist who started from Town R traveled for 2.2 hours at 40 mph:
\(\mathrm{Distance\,from\,Town\,R} = 40 × 2.2 = 88\) miles

Let's verify this makes sense: The motorist from Town S traveled \(55 × 2.2 = 121\) miles
Check: \(88 + 121 = 209\) miles ✓

Final Answer

The motorists are 88 miles from Town R when they pass each other.

Looking at our answer choices, this matches choice B: 88.

The answer is B.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding what distance needs to be found
Students often get confused about what the question is asking for. The question asks for the distance FROM TOWN R when they meet, but students might mistakenly think they need to find the distance from Town S, or the total distance traveled by both motorists combined. This leads to setting up the wrong equation or calculating the wrong final value.

Faltering Point 2: Not recognizing that both motorists travel for the same time
A key insight is that since both motorists start simultaneously and meet at the same point, they must have been traveling for exactly the same amount of time. Students sometimes miss this crucial relationship and try to set up separate time equations for each motorist, making the problem much more complicated than it needs to be.

Faltering Point 3: Incorrectly setting up the distance relationship
Students may fail to recognize that the sum of distances traveled by both motorists equals the total distance between towns (209 miles). Instead, they might try to subtract distances or create other incorrect relationships, leading to wrong equations from the start.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in division
When calculating t = 209 ÷ 95, students often make computational mistakes. The result is 2.2 hours, but students might incorrectly calculate this as 2.02 or round incorrectly to 2 hours, which would lead to wrong final answers.

Faltering Point 2: Using the wrong motorist's data for final calculation
After finding the meeting time (2.2 hours), students need to calculate how far the Town R motorist traveled (40 × 2.2 = 88 miles). However, students sometimes mistakenly use the Town S motorist's data (55 × 2.2 = 121 miles), giving them the distance from Town S instead of from Town R.

Faltering Point 3: Forgetting to verify the answer
Students might get the right setup and calculation but fail to check whether their answer makes sense. A quick verification (88 + 121 = 209) confirms the answer, but without this check, students might second-guess a correct answer or miss an arithmetic error.

Errors while selecting the answer

Faltering Point 1: Selecting the distance from the wrong starting point
If a student calculated both distances correctly (88 miles from Town R and 121 miles from Town S), they might select answer choice E (114) thinking it's close to 121, or get confused about which distance the question actually wants. The question specifically asks for distance FROM TOWN R, which is 88 miles.