A total of n trucks and cars are parked in a lot. If the number of cars is 1/4 the...

1. Translate the problem requirements: We need to clarify that "n" represents the total vehicles (trucks + cars), \(\text{cars} = \frac{1}{4} \times \text{trucks}\) means cars are one-fourth the number of trucks, and \(\frac{2}{3} \text{ of trucks are pickups}\) means pickups are two-thirds of all trucks. We need to find pickups as a fraction of the total n.
2. Express vehicle quantities in terms of trucks: Since cars depend on trucks \(\left(\frac{1}{4} \text{ relationship}\right)\), we'll use trucks as our base variable to express both cars and total vehicles, avoiding fractions within fractions.
3. Connect trucks to the total n: Use the relationship that total vehicles = trucks + cars to express trucks in terms of n, which will allow us to work with the given variable.
4. Calculate pickups as a fraction of total: Apply the \(\frac{2}{3} \text{ of trucks are pickups}\) relationship and express the final answer in terms of n by substituting our expression for trucks.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in simple terms:

• We have a parking lot with trucks and cars - the total number is called "n"
• The number of cars is \(\frac{1}{4}\) the number of trucks (so if there were 8 trucks, there would be 2 cars)
• \(\frac{2}{3}\) of the trucks are pickups (so if there were 9 trucks, 6 would be pickups)
• We need to find how many pickups there are, expressed as a fraction of the total vehicles n

Think of it this way: if you walked through the parking lot and counted all vehicles, you'd get n. We want to know what fraction of that n represents just the pickup trucks.

Process Skill: TRANSLATE - Converting the word relationships into mathematical understanding

2. Express vehicle quantities in terms of trucks

Since cars depend on the number of trucks, let's use trucks as our starting point. Let's call the number of trucks "T".

From the problem:

• Number of trucks = T
• Number of cars = \(\frac{1}{4} \times T\)
• Total vehicles = trucks + cars = \(T + \frac{1}{4}T\)

To add these together: \(T + \frac{1}{4}T = \frac{4}{4}T + \frac{1}{4}T = \frac{5}{4}T\)

So the total number of vehicles is \(\frac{5}{4}T\), which means our total \(n = \frac{5}{4}T\)

3. Connect trucks to the total n

Now we need to flip this relationship around. We know that:

\(n = \frac{5}{4}T\)

To find T in terms of n, we multiply both sides by \(\frac{4}{5}\):

\(T = n \times \frac{4}{5} = \frac{4}{5}n\)

Let's verify this makes sense: if \(n = 15\) total vehicles, then \(T = \frac{4}{5} \times 15 = 12\) trucks, and cars = \(\frac{1}{4} \times 12 = 3\) cars. Total: \(12 + 3 = 15\) ✓

4. Calculate pickups as a fraction of total

Now we can find the number of pickups. We know:

• Number of trucks = \(\frac{4}{5}n\)
• Pickups = \(\frac{2}{3}\) of the trucks

So: Number of pickups = \(\frac{2}{3} \times \frac{4}{5}n\)

Multiplying fractions: \(\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}\)

Therefore: Number of pickups = \(\frac{8}{15}n\)

Final Answer

The number of pickups in terms of n is \(\frac{8}{15}n\).

Looking at our answer choices, this matches choice D: \(\frac{8}{15}n\)

To double-check: If \(n = 15\) vehicles, we'd have 12 trucks and 3 cars. Of the 12 trucks, \(\frac{2}{3}\) would be pickups: \(\frac{2}{3} \times 12 = 8\) pickups. And indeed, \(\frac{8}{15} \times 15 = 8\) pickups ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship between cars and trucks

Students often confuse which variable depends on which. The problem states "the number of cars is \(\frac{1}{4}\) the number of trucks," but students might incorrectly interpret this as "the number of trucks is \(\frac{1}{4}\) the number of cars" or set up the equation backwards. This leads to expressing trucks in terms of cars instead of cars in terms of trucks, creating a fundamentally flawed approach.

2. Confusing "pickups" with "trucks"

Students frequently miss that pickups are a subset of trucks, not a separate category. They might treat the problem as having three distinct vehicle types (cars, trucks, pickups) instead of understanding that pickups are \(\frac{2}{3}\) of the trucks. This misunderstanding leads to incorrect total vehicle counts and wrong relationships.

3. Attempting to solve without establishing a clear variable relationship

Many students jump straight into calculations without first establishing what variable to use as their foundation. They might try to work directly with 'n' from the start instead of recognizing that expressing everything in terms of the number of trucks first makes the relationships clearer and the algebra more manageable.

Errors while executing the approach

1. Arithmetic errors when adding fractions

When calculating the total vehicles as \(T + \frac{1}{4}T\), students often make mistakes converting to common denominators. They might incorrectly calculate this as \(\frac{5}{4}T\) by adding \(1 + \frac{1}{4} = \frac{5}{4}\), but fail to properly convert the "1" to \(\frac{4}{4}\) first, leading to errors like getting \(\frac{2}{4}T\) or \(1.25T\) instead of \(\frac{5}{4}T\).

2. Incorrectly inverting the relationship to solve for T

When students have \(n = \frac{5}{4}T\) and need to find T in terms of n, they often make algebraic errors. Common mistakes include multiplying by \(\frac{5}{4}\) instead of \(\frac{4}{5}\), or incorrectly stating that \(T = \frac{5}{4}n\) instead of \(T = \frac{4}{5}n\). This error propagates through all subsequent calculations.

3. Errors in multiplying fractions

When calculating \(\frac{2}{3} \times \frac{4}{5}\), students frequently make multiplication errors. They might incorrectly multiply denominators and numerators in wrong combinations, getting results like \(\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}\), or make arithmetic errors like \(\frac{2 \times 4}{3 \times 5} = \frac{8}{8} = 1\), leading to completely wrong final expressions.

Errors while selecting the answer

1. Selecting the fraction that represents trucks instead of pickups

Students who correctly calculate that trucks represent \(\frac{4}{5}n\) of the total vehicles might accidentally select an answer choice that matches this intermediate result rather than continuing to find pickups. They might look for \(\frac{4}{5}n\) among the choices, and when not finding it exactly, select the closest option instead of completing the final step.

2. Confusing the calculated result with a different answer choice that looks similar

After correctly calculating \(\frac{8}{15}n\), students might misread the answer choices and select a fraction that looks similar, such as \(\frac{8}{12}n = \frac{2}{3}n\) if it were available, or misidentify which letter corresponds to \(\frac{8}{15}\). This is especially common when students are rushing or not double-checking their fraction matching.

Alternate Solutions

Smart Numbers Approach

Instead of working with variables algebraically, we can assign a specific numerical value for the number of trucks that makes our fraction calculations clean.

Step 1: Choose a convenient value for trucks

Since we need to work with fractions \(\frac{1}{4}\) (for cars) and \(\frac{2}{3}\) (for pickups), let's choose trucks = 12, because 12 is divisible by both 4 and 3, making our calculations clean.

Step 2: Calculate the number of cars

Cars = \(\frac{1}{4} \times \text{trucks} = \frac{1}{4} \times 12 = 3\) cars

Step 3: Find the total number of vehicles (n)

\(n = \text{trucks} + \text{cars} = 12 + 3 = 15\) total vehicles

Step 4: Calculate the number of pickups

Pickups = \(\frac{2}{3} \times \text{trucks} = \frac{2}{3} \times 12 = 8\) pickups

Step 5: Express pickups as a fraction of total vehicles

Fraction of pickups = \(\frac{8}{15}\) of the total vehicles

Therefore, pickups = \(\frac{8}{15} \times n\)

This matches answer choice D: \(\frac{8}{15}n\)