A certain airline's fleet consisted of 60 Type A planes at the beginning of 1980. At the end of each...

1. Translate the problem requirements: We start with 60 Type A planes in 1980. Each year we lose 3 Type A planes and gain 4 Type B planes. We need to find when Type A planes become less than 50% of the total fleet.
2. Set up the yearly tracking framework: Establish how to calculate Type A planes, Type B planes, and total fleet for any given year after the changes.
3. Determine the threshold condition: Figure out when Type A planes will be less than 50% of the total fleet by setting up the inequality.
4. Calculate year by year until threshold is met: Track the fleet composition each year until Type A planes drop below 50% of the total.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what's happening in plain English: We have an airline that starts with 60 Type A planes at the beginning of 1980. Every year, they make the same changes - they get rid of 3 Type A planes (retirement) and they buy 4 new Type B planes. We want to know how many years it takes until the Type A planes make up less than half of all the planes they own.

To be very clear about what "less than 50 percent" means: if Type A planes are exactly 50% of the fleet, that's NOT what we want - we need them to be LESS than 50%. So if the total fleet has 100 planes, we need fewer than 50 Type A planes.

Process Skill: TRANSLATE

2. Set up the yearly tracking framework

Let's figure out how to calculate what happens each year. We'll use simple patterns:

After n years:

- Type A planes = \(60 - 3\mathrm{n}\) (we started with 60 and lose 3 each year)

- Type B planes = \(4\mathrm{n}\) (we started with 0 and gain 4 each year)

- Total fleet = Type A + Type B = \((60 - 3\mathrm{n}) + 4\mathrm{n} = 60 + \mathrm{n}\)

This makes sense: each year we lose 3 planes but gain 4 planes, so our total fleet grows by 1 plane per year.

3. Determine the threshold condition

We want Type A planes to be less than 50% of the total fleet. In mathematical terms:

Type A planes < 50% of total fleet

\(60 - 3\mathrm{n} < 0.5 \times (60 + \mathrm{n})\)

\(60 - 3\mathrm{n} < 30 + 0.5\mathrm{n}\)

\(60 - 30 < 3\mathrm{n} + 0.5\mathrm{n}\)

\(30 < 3.5\mathrm{n}\)

\(\mathrm{n} > 30/3.5 = 60/7 \approx 8.57\)

Since n must be a whole number of years, we need n ≥ 9 years.

Process Skill: INFER

4. Calculate year by year until threshold is met

Let's verify our answer by tracking the actual numbers year by year:

Year 0 (beginning of 1980): 60 Type A, 0 Type B, Total = 60
Type A percentage = \(60/60 = 100\%\)

Year 1 (end of 1980): 57 Type A, 4 Type B, Total = 61
Type A percentage = \(57/61 \approx 93.4\%\)

Year 6: 42 Type A, 24 Type B, Total = 66
Type A percentage = \(42/66 \approx 63.6\%\)

Year 7: 39 Type A, 28 Type B, Total = 67
Type A percentage = \(39/67 \approx 58.2\%\)

Year 8: 36 Type A, 32 Type B, Total = 68
Type A percentage = \(36/68 \approx 52.9\%\)

Year 9: 33 Type A, 36 Type B, Total = 69
Type A percentage = \(33/69 \approx 47.8\%\)

At year 9, Type A planes make up about 47.8% of the fleet, which is less than 50% for the first time.

4. Final Answer

The answer is D. 9 years.

At the end of the 9th year, Type A planes represent approximately 47.8% of the total fleet, which is the first time they drop below 50% of the total fleet.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "less than 50 percent" requirement

Students often confuse "less than 50%" with "less than or equal to 50%" or "at most 50%". They might think that exactly 50% satisfies the condition, when the problem specifically asks for the first time Type A planes become LESS than half the fleet. This leads to selecting an answer that's one year too early.

2. Confusion about timing - beginning vs. end of year

The problem states the fleet "consisted of 60 Type A planes at the beginning of 1980" and changes happen "at the end of each year, starting with 1980". Students may get confused about whether to count from the beginning or end of 1980, or may miscount the years by starting from year 1 instead of year 0.

3. Setting up incorrect expressions for fleet composition

Some students struggle to correctly model how the fleet changes each year. They might incorrectly account for the retirement and acquisition, such as thinking both Type A retirement and Type B acquisition affect the Type A count, or forgetting that the total fleet size changes each year.

Errors while executing the approach

1. Arithmetic mistakes in solving the inequality

When solving \(60 - 3\mathrm{n} < 0.5(60 + \mathrm{n})\), students commonly make errors like: incorrectly distributing 0.5, making sign errors when moving terms across the inequality, or miscalculating \(30/3.5 = 60/7 \approx 8.57\).

2. Calculation errors in year-by-year tracking

If using the verification approach, students may make simple arithmetic errors when calculating the number of planes each year (60-3n for Type A, 4n for Type B) or when computing percentages (like calculating \(33/69\) incorrectly).

3. Incorrect inequality direction interpretation

After finding n > 8.57, students might incorrectly conclude that n = 8 is sufficient, forgetting that since we need n to be greater than 8.57, we must round UP to n = 9 years.

Errors while selecting the answer

1. Off-by-one error due to rounding confusion

Students correctly find n > 8.57 but select 8 instead of 9, either because they round down instead of up, or because they think "more than 8 and a bit" means 8 is close enough.

2. Misreading the percentage threshold

Even after correct calculations, students might select the year when Type A planes are closest to 50% rather than the first year they drop below 50%, leading to selecting year 8 (where Type A is about 52.9%) instead of year 9 (where Type A is about 47.8%).