According to a car dealer's sales report, 1/3 of the cars sold during a certain period were sedans and 1/5...

1. Translate the problem requirements: We need to clarify what "\(\frac{1}{5}\) of the other cars" means - this refers to \(\frac{1}{5}\) of the cars that are NOT sedans, and find the relationship between sedans and station wagons in terms of N.
2. Express everything in terms of total cars: Set up the total number of cars as our base variable and express both sedans and station wagons as fractions of this total.
3. Establish the station wagon relationship: Use the fact that station wagons are \(\frac{1}{5}\) of non-sedan cars to create an equation involving N and the total cars.
4. Solve for total cars in terms of N: Manipulate the station wagon equation to express the total number of cars in terms of N.
5. Calculate sedans using the total: Apply the \(\frac{1}{3}\) relationship to find the number of sedans in terms of N.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is telling us in everyday language. We have a car dealer who sold some cars during a certain period. Out of all the cars sold:

• \(\frac{1}{3}\) of ALL cars sold were sedans
• \(\frac{1}{5}\) of the OTHER cars (meaning the cars that were NOT sedans) were station wagons

The key insight here is understanding what "other cars" means - it's specifically referring to the cars that aren't sedans. So if sedans make up \(\frac{1}{3}\) of all cars, then the "other cars" make up the remaining \(\frac{2}{3}\) of all cars.

Process Skill: TRANSLATE - Converting the phrase "\(\frac{1}{5}\) of the other cars" into mathematical understanding is crucial for setting up the problem correctly.

2. Express everything in terms of total cars

Let's call the total number of cars sold "T". Now we can express everything clearly:

• Sedans = \(\frac{1}{3}\) of total cars = \(\frac{\mathrm{T}}{3}\)
• Non-sedan cars = \(\frac{2}{3}\) of total cars = \(\frac{2\mathrm{T}}{3}\)
• Station wagons = \(\frac{1}{5}\) of non-sedan cars = \(\frac{1}{5} \times \frac{2\mathrm{T}}{3} = \frac{2\mathrm{T}}{15}\)

This gives us a clear picture: out of every 15 cars sold, 5 are sedans, 10 are non-sedans, and 2 of those non-sedans are station wagons.

3. Establish the station wagon relationship

Now we know that the number of station wagons sold is \(\frac{2\mathrm{T}}{15}\), but the problem tells us that N station wagons were sold. This means:

\(\mathrm{N} = \frac{2\mathrm{T}}{15}\)

This equation connects the number of station wagons (N) to the total number of cars (T). We'll use this to find T in terms of N.

4. Solve for total cars in terms of N

Starting with \(\mathrm{N} = \frac{2\mathrm{T}}{15}\), we need to solve for T:

\(\mathrm{N} = \frac{2\mathrm{T}}{15}\)

Multiply both sides by 15: \(15\mathrm{N} = 2\mathrm{T}\)

Divide both sides by 2: \(\mathrm{T} = \frac{15\mathrm{N}}{2}\)

So the total number of cars sold is \(\frac{15\mathrm{N}}{2}\).

5. Calculate sedans using the total

We know that sedans make up \(\frac{1}{3}\) of all cars sold, and we just found that the total is \(\frac{15\mathrm{N}}{2}\). Therefore:

Number of sedans = \(\frac{1}{3} \times \mathrm{T} = \frac{1}{3} \times \frac{15\mathrm{N}}{2} = \frac{15\mathrm{N}}{6} = \frac{5\mathrm{N}}{2}\)

Let's verify this makes sense: if \(\frac{5\mathrm{N}}{2}\) sedans and N station wagons were sold, and station wagons are \(\frac{1}{5}\) of non-sedans, then non-sedans should be 5N. Indeed, total cars (\(\frac{15\mathrm{N}}{2}\)) minus sedans (\(\frac{5\mathrm{N}}{2}\)) equals \(\frac{10\mathrm{N}}{2} = 5\mathrm{N}\) non-sedans. And \(\frac{1}{5}\) of 5N equals N station wagons. ✓

4. Final Answer

The number of sedans sold is \(\frac{5\mathrm{N}}{2}\), which matches answer choice D.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "other cars"
Students often misunderstand what "\(\frac{1}{5}\) of the other cars sold were station wagons" means. They might think "other cars" refers to all remaining car types besides station wagons, rather than understanding it means "non-sedan cars." This fundamental misinterpretation leads to setting up incorrect equations from the start.

Faltering Point 2: Confusing the fraction base
Students may incorrectly think that station wagons are \(\frac{1}{5}\) of ALL cars sold, rather than \(\frac{1}{5}\) of the non-sedan cars. This happens when they don't carefully track what each fraction is "of" - missing that the station wagon fraction has a different base (\(\frac{2}{3}\) of total cars) than the sedan fraction (total cars).

Faltering Point 3: Working backwards incorrectly
Some students struggle with the reverse-engineering aspect of the problem. They know the final count (N station wagons) and need to work backwards to find sedans, but they may try to work forward from an assumed total instead of using the given information about station wagons to establish the total.

Errors while executing the approach

Faltering Point 1: Fraction multiplication errors
When calculating that station wagons = \(\frac{1}{5} \times \frac{2}{3} \times \mathrm{T} = \frac{2\mathrm{T}}{15}\), students often make arithmetic mistakes with fraction multiplication. They might get \(\frac{2\mathrm{T}}{8}\) or \(\frac{1\mathrm{T}}{15}\), leading to incorrect relationships between N and T.

Faltering Point 2: Algebraic manipulation mistakes
When solving \(\mathrm{N} = \frac{2\mathrm{T}}{15}\) for T, students may incorrectly get \(\mathrm{T} = \frac{2\mathrm{N}}{15}\) instead of \(\mathrm{T} = \frac{15\mathrm{N}}{2}\). This happens when they confuse which variable to multiply or divide by which number, especially when dealing with fractions.

Errors while selecting the answer

No likely faltering points

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart value for N
Let's choose N = 4 station wagons (this works well with the fraction \(\frac{1}{5}\) in our problem).

Step 2: Find the number of non-sedan cars
Since station wagons represent \(\frac{1}{5}\) of the "other cars" (non-sedans):
If 4 station wagons = \(\frac{1}{5}\) of non-sedan cars
Then non-sedan cars = \(4 \times 5 = 20\) cars

Step 3: Find the total number of cars
If \(\frac{1}{3}\) of all cars are sedans, then \(\frac{2}{3}\) of all cars are non-sedans.
Since non-sedans = 20 cars = \(\frac{2}{3}\) of total cars
Total cars = \(20 \div \frac{2}{3} = 20 \times \frac{3}{2} = 30\) cars

Step 4: Calculate the number of sedans
Sedans = \(\frac{1}{3}\) of total cars = \(\frac{1}{3} \times 30 = 10\) sedans

Step 5: Express sedans in terms of N
When N = 4, sedans = 10
Therefore: sedans = \(\frac{10}{4} \times \mathrm{N} = \frac{5}{2} \times \mathrm{N} = \frac{5\mathrm{N}}{2}\)

Step 6: Verify with answer choices
Our result \(\frac{5\mathrm{N}}{2}\) matches answer choice D.

Verification with another smart number:
Let's try N = 2:
• Non-sedan cars = \(2 \times 5 = 10\)
• Total cars = \(10 \div \frac{2}{3} = 15\)
• Sedans = \(\frac{15}{3} = 5\)
• Formula check: \(\frac{5\mathrm{N}}{2} = \frac{5(2)}{2} = 5\) ✓