Last year, a certain public transportation system sold an average (arithmetic mean) of 41,000 tickets per day on weekdays (Monday...

1. Translate the problem requirements: We need to find the total tickets sold in one year. This means calculating tickets sold on all weekdays (Monday-Friday) plus all weekend days (Saturday-Sunday) for the entire year.
2. Determine the weekly ticket pattern: Calculate how many tickets are sold in one typical week by finding weekday total (\(5 \text{ days} \times 41,000\)) plus weekend total (\(2 \text{ days} \times 18,000\)).
3. Scale to annual volume: Multiply the weekly total by 52 weeks to get the approximate annual total, since the answer choices are far apart and allow for reasonable approximation.
4. Match to closest answer choice: Compare our calculated result with the given options to identify the best match.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're looking for in everyday terms. We want to know how many tickets this transportation system sold in an entire year.

Think of it like this: imagine you're tracking ticket sales for a bus or subway system. Some days are busier than others. Weekdays (Monday through Friday) are busy because people go to work and school. Weekends (Saturday and Sunday) are quieter because fewer people need to commute.

We're told:

- Weekdays average 41,000 tickets per day
- Weekend days average 18,000 tickets per day

To find the yearly total, we need to add up all the weekday sales and all the weekend sales for the entire year.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Determine the weekly ticket pattern

Now let's figure out how many tickets are sold in one typical week. A week has 7 days: 5 weekdays and 2 weekend days.

Weekday tickets in one week:
\(5 \text{ weekdays} \times 41,000 \text{ tickets per weekday} = 205,000 \text{ tickets}\)

Weekend tickets in one week:
\(2 \text{ weekend days} \times 18,000 \text{ tickets per weekend day} = 36,000 \text{ tickets}\)

Total tickets in one week:
\(205,000 + 36,000 = 241,000 \text{ tickets per week}\)

This gives us a clear pattern: every week, the system sells approximately 241,000 tickets.

3. Scale to annual volume

Now we need to go from weekly sales to yearly sales. There are 52 weeks in a year, so:

Total yearly tickets = \(241,000 \text{ tickets per week} \times 52 \text{ weeks}\)

Let's calculate this step by step:
\(241,000 \times 52\)

To make this easier, let's break it down:
\(241,000 \times 50 = 12,050,000\)
\(241,000 \times 2 = 482,000\)

Total: \(12,050,000 + 482,000 = 12,532,000 \text{ tickets}\)

So approximately 12.5 million tickets were sold last year.

4. Match to closest answer choice

Our calculation gives us approximately 12,532,000 tickets, which is about 12.5 million.

Looking at our answer choices:

1. 1 million - too small
2. 1.25 million - too small
3. 10 million - close but still too small
4. 12.5 million - matches our calculation perfectly
5. 125 million - way too large

Final Answer

The answer is (D) 12.5 million tickets. Our systematic calculation of weekly sales (241,000) multiplied by 52 weeks gives us approximately 12.5 million tickets sold last year.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the time frame structure

Students may incorrectly assume that all days in a year are the same, missing that weekdays and weekends have different ticket sales patterns. They might try to use just one average (like 41,000) for all 365 days, leading to a completely wrong calculation.

2. Confusing daily averages with weekly totals

Some students might misinterpret the given information and think that 41,000 is the total for all weekdays combined in a week, rather than the daily average for each individual weekday. This would lead them to use \(41,000 + 18,000 = 59,000\) per week instead of calculating properly.

Errors while executing the approach

1. Arithmetic mistakes in multiplication

When calculating \(241,000 \times 52\), students often make computational errors. They might incorrectly calculate \(241,000 \times 50\) or mess up the addition of \(12,050,000 + 482,000\), leading to answers that are off by significant amounts.

2. Using 365 days instead of 52 weeks

Students might attempt to calculate weekdays and weekend days separately for the entire year (like 260 weekdays and 104 weekend days) instead of using the simpler weekly pattern approach. While this method can work, it's more prone to counting errors and arithmetic mistakes.

Errors while selecting the answer

1. Misreading the scale of answer choices

Students might calculate correctly to get approximately 12,532,000 but then select choice (C) 10 million because they misread their own calculation or don't properly convert between numerical form and 'millions.' They might think 12.5 million looks too large compared to their mental estimate.