Last week the ratio of the number of trucks sold to the number of cars sold by a certain dealership...

1. Translate the problem requirements: We have a ratio of trucks to cars (\(7:9\)) and know that 4 more cars than trucks were sold. We need to find the actual number of trucks sold.
2. Set up the ratio relationship: Use the given ratio to express the actual quantities in terms of a common multiplier.
3. Apply the numerical constraint: Use the fact that cars exceed trucks by exactly 4 to create an equation.
4. Solve for the scaling factor: Calculate the multiplier that makes the ratio match the real-world constraint.
5. Calculate the final answer: Apply the scaling factor to find the actual number of trucks sold.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• The dealership sold trucks and cars in a specific proportion: for every 7 trucks sold, 9 cars were sold
• We also know that exactly 4 more cars than trucks were sold
• We need to find how many trucks were actually sold

Think of it this way: if we imagine the ratio \(7:9\) as a recipe, we're looking for how many times this "recipe" was used to get the actual sales numbers.

Process Skill: TRANSLATE - Converting the ratio relationship and constraint into mathematical understanding

2. Set up the ratio relationship

Since the ratio of trucks to cars is \(7:9\), we can say:

• If we multiply both parts of the ratio by the same number, we get the actual sales
• Let's call this multiplier "k"
• So trucks sold = \(7k\) and cars sold = \(9k\)

This makes sense because ratios tell us about proportional relationships. If the basic pattern is 7 trucks for every 9 cars, then the actual numbers must be some multiple of this pattern.

Mathematically: Trucks = \(7k\), Cars = \(9k\)

3. Apply the numerical constraint

Now we use the fact that 4 more cars than trucks were sold.
In everyday language: Cars sold - Trucks sold = 4

Substituting our expressions:
\(9k - 7k = 4\)
\(2k = 4\)

This equation captures the key constraint that links our ratio to the real numbers.

Process Skill: APPLY CONSTRAINTS - Using the numerical difference to create a solvable equation

4. Solve for the scaling factor

From our equation \(2k = 4\):
\(k = 4 ÷ 2 = 2\)

This tells us that the basic ratio pattern (\(7:9\)) was repeated exactly 2 times to get our actual sales numbers.

5. Calculate the final answer

Now we can find the actual number of trucks sold:
Trucks sold = \(7k = 7 × 2 = 14\)

Let's verify this makes sense:

• Cars sold = \(9k = 9 × 2 = 18\)
• Difference: \(18 - 14 = 4\) ✓
• Ratio: \(14:18 = 7:9\) ✓

Final Answer

The dealership sold 14 trucks last week.

This matches answer choice C.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the ratio relationship

Students often struggle to understand that when given a ratio like \(7:9\), the actual numbers must be multiples of these values (\(7k\) and \(9k\)), not exactly 7 and 9. They might try to work directly with 7 trucks and 9 cars, leading to confusion when applying the constraint that 4 more cars than trucks were sold.

2. Incorrectly setting up the constraint equation

Some students reverse the constraint and write "trucks - cars = 4" instead of "cars - trucks = 4". This happens because they might focus on the word "trucks" appearing first in the sentence "4 more cars than trucks were sold" and incorrectly interpret the relationship.

3. Attempting to solve without using variables

Students might try to guess and check with the answer choices instead of setting up the systematic algebraic approach with the scaling factor k. This leads to inefficient problem-solving and potential errors in verification.

Errors while executing the approach

1. Arithmetic errors in solving the equation

When solving \(2k = 4\), students might make simple calculation errors, such as getting \(k = 8\) instead of \(k = 2\), or incorrectly simplifying \(9k - 7k\) as something other than \(2k\).

2. Using the wrong value to calculate the final answer

After finding \(k = 2\), students might calculate the number of cars (\(9k = 18\)) instead of trucks (\(7k = 14\)), especially if they lose track of what the question is actually asking for.

Errors while selecting the answer

1. Failing to verify the answer

Students might arrive at the correct calculation but fail to double-check that their answer satisfies both conditions (the \(7:9\) ratio and the difference of 4). This verification step is crucial for catching earlier errors and building confidence in the solution.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers based on the ratio

Since the ratio of trucks to cars is \(7:9\), let's use the most natural choice:

• Number of trucks = \(7k\) (where k is our multiplier)
• Number of cars = \(9k\) (using the same multiplier to maintain the \(7:9\) ratio)

Step 2: Apply the constraint about the difference

We know that 4 more cars than trucks were sold:

Cars - Trucks = 4

\(9k - 7k = 4\)

\(2k = 4\)

\(k = 2\)

Step 3: Calculate the actual numbers

Now that we know \(k = 2\):

• Number of trucks = \(7k = 7(2) = 14\)
• Number of cars = \(9k = 9(2) = 18\)

Step 4: Verify our answer

• Ratio check: \(14:18 = 7:9\) ✓
• Difference check: \(18 - 14 = 4\) ✓

Therefore, 14 trucks were sold.