Of the vehicles (cars and trucks) for sale at a certain dealership, 48 percent are cars that have 4 doors....

1. Translate the problem requirements: We need to understand that vehicles = cars + trucks, and we're given information about 4-door cars as a percentage of all vehicles and as a percentage of all cars. We need to find cars without 4 doors as a percentage of all vehicles.
2. Find the percentage of cars among all vehicles: Use the given information about 4-door cars to determine what percentage of all vehicles are cars.
3. Calculate cars without 4 doors as percentage of all cars: Since 80% of cars have 4 doors, the remaining percentage of cars don't have 4 doors.
4. Convert to percentage of all vehicles: Apply the percentage of cars without 4 doors to the total percentage of cars among all vehicles to get the final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language. Think of this dealership as having a parking lot with cars and trucks. We're told that when we look at ALL vehicles (cars AND trucks together), 48% of them are cars that happen to have 4 doors.

We're also told that if we focus just on the cars (ignoring trucks completely), 80% of those cars have 4 doors.

Our goal is to find what percentage of ALL vehicles are cars that do NOT have 4 doors.

To organize our thinking:

• Total vehicles = Cars + Trucks
• 4-door cars = 48% of all vehicles
• 4-door cars = 80% of all cars
• We want: Cars without 4 doors as % of all vehicles

Process Skill: TRANSLATE

2. Find the percentage of cars among all vehicles

Here's the key insight: If 4-door cars make up 48% of all vehicles, and 4-door cars represent 80% of all cars, we can figure out what percentage of all vehicles are cars.

Think of it this way: If 80% of cars have 4 doors, and those 4-door cars represent 48% of all vehicles, then we can work backwards.

Let's say cars make up X% of all vehicles.
Then: 80% of X% = 48% of all vehicles
In other words: \(0.80 \times \mathrm{X} = 48\)
Solving: \(\mathrm{X} = 48 \div 0.80 = 60\)

So cars make up 60% of all vehicles at the dealership.

3. Calculate cars without 4 doors as percentage of all cars

This step is straightforward. If 80% of cars have 4 doors, then the remaining cars do not have 4 doors.

Cars without 4 doors = \(100\% - 80\% = 20\%\) of all cars

4. Convert to percentage of all vehicles

Now we combine our findings:

• Cars represent 60% of all vehicles
• Cars without 4 doors represent 20% of all cars

To find cars without 4 doors as a percentage of all vehicles:
\(20\% \text{ of } 60\% = 0.20 \times 60\% = 12\%\)

Therefore, 12% of all vehicles are cars that do not have 4 doors.

Final Answer

The answer is 12%, which corresponds to choice (A).

To verify:

• 4-door cars: 48% of all vehicles ✓
• Cars without 4 doors: 12% of all vehicles
• Total cars: \(48\% + 12\% = 60\%\) of all vehicles
• \(80\% \text{ of } 60\% = 48\%\) (matches our given information) ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what "48 percent are cars that have 4 doors" means

Students often confuse this statement and think it means "48% of cars have 4 doors" instead of "48% of ALL vehicles are 4-door cars." This fundamental misreading completely derails the solution approach since they would then incorrectly assume that 48% and 80% both refer to percentages of cars only.

2. Failing to recognize the need to find total percentage of cars first

Many students jump directly to trying to calculate non-4-door cars without realizing they first need to determine what percentage of all vehicles are cars. They miss that the key insight is using the relationship between "80% of cars have 4 doors" and "4-door cars are 48% of all vehicles" to work backwards and find the total car percentage.

3. Confusing the reference groups (cars vs. all vehicles)

Students struggle to keep track of whether a percentage refers to "percentage of cars" or "percentage of all vehicles." This leads to incorrect setups where they mix these reference groups inappropriately, such as trying to directly subtract 48% from 80% without proper conversion.

Errors while executing the approach

1. Arithmetic errors in the division step

When solving \(48 \div 0.80 = 60\), students commonly make calculation mistakes, getting answers like 38.4 \((48 \times 0.8)\) or 600 (forgetting decimal placement). These errors propagate through the rest of the solution.

2. Incorrect percentage calculations

Students often confuse when to multiply by 100 or divide by 100 when converting between decimals and percentages. For example, they might calculate \(0.20 \times 60 = 12\) but then incorrectly think they need to convert this to 1200% or 1.2%.

3. Using wrong formula setup

Instead of correctly setting up \(0.80 \times \mathrm{X} = 48\), students might write \(0.48 \times \mathrm{X} = 80\) or other incorrect relationships, leading to wrong values for the total percentage of cars.

Errors while selecting the answer

1. Selecting 20% instead of 12%

Students correctly find that 20% of cars don't have 4 doors, but forget the final conversion step and select 20% as their answer instead of recognizing this needs to be converted to a percentage of all vehicles (which gives 12%).

2. Choosing 60% based on intermediate calculation

After correctly calculating that cars make up 60% of all vehicles, some students mistakenly select this as their final answer, forgetting that the question asks specifically for cars WITHOUT 4 doors as a percentage of all vehicles.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for total vehicles

Let's say there are 100 total vehicles at the dealership. This makes percentage calculations straightforward since each vehicle represents 1%.

Step 2: Calculate number of 4-door cars

We're told that 48% of all vehicles are cars with 4 doors.
Number of 4-door cars = \(48\% \times 100 = 48\) cars

Step 3: Calculate total number of cars

We know that 80% of all cars have 4 doors, and we just found there are 48 cars with 4 doors.
If 80% of cars = 48 cars, then:
Total cars = \(48 \div 0.80 = 60\) cars

Step 4: Calculate cars without 4 doors

Cars without 4 doors = Total cars - Cars with 4 doors
Cars without 4 doors = \(60 - 48 = 12\) cars

Step 5: Convert to percentage of all vehicles

Since we started with 100 total vehicles, and 12 cars don't have 4 doors:
Percentage = \(\frac{12}{100} = 12\%\)

Verification:

• Total vehicles: 100
• Cars with 4 doors: 48 (48% of vehicles ✓)
• Total cars: 60
• \(80\% \text{ of } 60\) cars = 48 cars with 4 doors ✓
• Cars without 4 doors: 12 (12% of vehicles)