- Translate the problem requirements: We need to find how many passengers boarded at the first stop, given that the average across the first 9 stops is 30 passengers, and each subsequent stop has 5 fewer passengers than the previous stop.
- Set up the passenger pattern using the decreasing sequence: Express the number of passengers at each of the first 9 stops in terms of the first stop value, using the "5 fewer" constraint.
- Apply the average constraint to create an equation: Use the fact that the sum of passengers across 9 stops divided by 9 equals 30 to set up a solvable equation.
- Solve for the first stop value: Calculate the number of passengers who boarded at the first stop and verify against answer choices.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in everyday terms:
- A train makes 11 stops total, but we only care about the first 9 stops for our calculation
- On average, 30 passengers board at each of these first 9 stops
- Starting from the second stop, each stop has exactly 5 fewer passengers boarding than the stop before it
- We need to find how many passengers boarded at the very first stop
Think of this like a staircase going down - each step down represents 5 fewer passengers than the step before.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
2. Set up the passenger pattern using the decreasing sequence
Let's call the number of passengers at the first stop \(\mathrm{x}\). Since each subsequent stop has 5 fewer passengers, we can write out the pattern:
- Stop 1: \(\mathrm{x}\) passengers
- Stop 2: \((\mathrm{x} - 5)\) passengers
- Stop 3: \((\mathrm{x} - 10)\) passengers
- Stop 4: \((\mathrm{x} - 15)\) passengers
- Stop 5: \((\mathrm{x} - 20)\) passengers
- Stop 6: \((\mathrm{x} - 25)\) passengers
- Stop 7: \((\mathrm{x} - 30)\) passengers
- Stop 8: \((\mathrm{x} - 35)\) passengers
- Stop 9: \((\mathrm{x} - 40)\) passengers
Notice the pattern: each stop subtracts an additional 5 from the original amount. This creates what mathematicians call an arithmetic sequence with first term \(\mathrm{x}\) and common difference \(-5\).
3. Apply the average constraint to create an equation
We know that the average of these 9 stops is 30 passengers. In plain English, this means:
(Total passengers across all 9 stops) ÷ 9 = 30
Therefore: Total passengers across all 9 stops = \(30 × 9 = 270\)
Now let's add up all the passengers from our pattern:
\(\mathrm{x} + (\mathrm{x}-5) + (\mathrm{x}-10) + (\mathrm{x}-15) + (\mathrm{x}-20) + (\mathrm{x}-25) + (\mathrm{x}-30) + (\mathrm{x}-35) + (\mathrm{x}-40) = 270\)
Simplifying the left side:
- We have 9 copies of \(\mathrm{x}\): \(9\mathrm{x}\)
- We subtract: \(5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 = 180\)
So our equation becomes: \(9\mathrm{x} - 180 = 270\)
Process Skill: MANIPULATE - Converting the average constraint into a solvable equation
4. Solve for the first stop value
Now we solve our equation step by step:
\(9\mathrm{x} - 180 = 270\)
Add 180 to both sides:
\(9\mathrm{x} = 270 + 180\)
\(9\mathrm{x} = 450\)
Divide both sides by 9:
\(\mathrm{x} = 450 ÷ 9 = 50\)
Let's verify this makes sense by checking a few values:
- Stop 1: 50 passengers
- Stop 2: 45 passengers
- Stop 3: 40 passengers
- Stop 9: \(50 - 40 = 10\) passengers
Average = \((50 + 45 + 40 + 35 + 30 + 25 + 20 + 15 + 10) ÷ 9 = 270 ÷ 9 = 30\) ✓
Final Answer
50 passengers boarded the train at its first stop.
Looking at our answer choices, this corresponds to Answer Choice B: 50, which matches the given correct answer.
Common Faltering Points
Errors while devising the approach
- Misinterpreting the scope of the average: Students may think the average of 30 passengers applies to all 11 stops instead of just the first 9 stops. This leads them to set up the wrong equation with 11 terms instead of 9, completely throwing off their calculation.
- Confusing the decreasing pattern: Students might misread 'at each stop, beginning with the second, 5 fewer passengers boarded than at the previous stop' and think that 5 fewer passengers board compared to some fixed number (like 30) rather than understanding it's a sequential decrease where each stop has 5 fewer than the immediately previous stop.
- Setting up the wrong variable: Instead of letting \(\mathrm{x}\) represent the number of passengers at the first stop, students might let \(\mathrm{x}\) represent the average (30) or try to work backwards from the last stop, making the problem unnecessarily complicated.
Errors while executing the approach
- Arithmetic errors in the sequence: When writing out the decreasing pattern (\(\mathrm{x}, \mathrm{x}-5, \mathrm{x}-10, \mathrm{x}-15, ..., \mathrm{x}-40\)), students often make mistakes in the subtraction amounts, perhaps writing \(\mathrm{x}-5, \mathrm{x}-5, \mathrm{x}-5\) instead of the cumulative decreases, or miscounting which stop corresponds to which decrease.
- Calculation errors when summing the sequence: Students frequently make mistakes when adding up the terms \(5+10+15+20+25+30+35+40\), either by missing terms, adding incorrectly, or not recognizing this as the sum of an arithmetic sequence that could be calculated more efficiently.
- Algebraic manipulation errors: When solving \(9\mathrm{x} - 180 = 270\), students might forget to add 180 to both sides, make sign errors, or divide incorrectly, leading to answers that don't match any of the given choices.
Errors while selecting the answer
No likely faltering points - once students correctly solve for \(\mathrm{x} = 50\), the answer clearly matches choice B, and there's no additional interpretation or conversion required.
