During the four years that Mrs. Lopez owned her car, she found that her total car expenses were $18,000. Fuel...

1. Translate the problem requirements: Break down the \(\$18,000\) total car expenses into categories: fuel/maintenance (\(\frac{1}{3}\) of total), depreciation (\(\frac{3}{5}\) of remainder after fuel/maintenance), insurance and financing together (\(\frac{1}{5}\) of total, with insurance = \(3 \times\) financing), and taxes/license fees (whatever remains). Find the difference between financing costs and taxes/license fees.
2. Calculate fuel/maintenance and remaining amount: Find \(\frac{1}{3}\) of \(\$18,000\) for fuel/maintenance, then determine what's left for other expenses.
3. Determine depreciation from the remainder: Calculate \(\frac{3}{5}\) of the amount remaining after fuel/maintenance to find depreciation costs.
4. Find individual insurance and financing costs: Use the constraint that insurance + financing = \(\frac{1}{5}\) of total and insurance = \(3 \times\) financing to solve for each.
5. Calculate taxes/license fees and compare: Find taxes/license fees as the remaining unaccounted amount, then determine the difference with financing costs.

Execution of Strategic Approach

1. Translate the problem requirements

Let me break down what Mrs. Lopez spent her \(\$18,000\) on in plain English:

• Fuel and maintenance took up \(\frac{1}{3}\) of her total expenses
• After paying for fuel and maintenance, she had some money left over. Depreciation ate up \(\frac{3}{5}\) of that leftover amount
• Insurance and financing together cost \(\frac{1}{5}\) of the total \(\$18,000\), and insurance cost exactly 3 times what financing cost
• Everything else went to taxes and license fees

We need to find how much more (or less) she spent on financing compared to taxes and license fees.

Process Skill: TRANSLATE - Converting the fractional relationships into a clear expense breakdown

2. Calculate fuel/maintenance and remaining amount

Let's start with fuel and maintenance costs:

Fuel and maintenance = \(\frac{1}{3}\) of \(\$18,000 = \$18,000 \div 3 = \$6,000\)

After paying for fuel and maintenance, Mrs. Lopez had:

Remaining amount = \(\$18,000 - \$6,000 = \$12,000\)

This \(\$12,000\) is what she had left to cover depreciation, insurance, financing, and taxes/license fees.

3. Determine depreciation from the remainder

Depreciation took up \(\frac{3}{5}\) of that remaining \(\$12,000\):

Depreciation = \(\frac{3}{5} \times \$12,000 = 3 \times \$2,400 = \$7,200\)

So far we've accounted for:

• Fuel/maintenance: \(\$6,000\)
• Depreciation: \(\$7,200\)
• Total so far: \(\$6,000 + \$7,200 = \$13,200\)

4. Find individual insurance and financing costs

Insurance and financing together cost \(\frac{1}{5}\) of the total:

Insurance + Financing = \(\frac{1}{5} \times \$18,000 = \$3,600\)

Since insurance costs 3 times what financing costs:

Insurance = \(3 \times\) Financing

Substituting into our equation:

\(3 \times\) Financing + Financing = \(\$3,600\)
\(4 \times\) Financing = \(\$3,600\)
Financing = \(\$3,600 \div 4 = \$900\)

Therefore:

Insurance = \(3 \times \$900 = \$2,700\)

Let's verify: \(\$2,700 + \$900 = \$3,600\) ✓

5. Calculate taxes/license fees and compare

Now let's find how much went to taxes and license fees:

Total expenses so far:

• Fuel/maintenance: \(\$6,000\)
• Depreciation: \(\$7,200\)
• Insurance: \(\$2,700\)
• Financing: \(\$900\)
• Subtotal: \(\$16,800\)

Taxes and license fees = \(\$18,000 - \$16,800 = \$1,200\)

Finally, let's compare financing costs to taxes/license fees:

Financing cost: \(\$900\)
Taxes/license fees: \(\$1,200\)
Difference: \(\$900 - \$1,200 = -\$300\)

This means financing cost \(\$300\) LESS than taxes and license fees.

4. Final Answer

The cost of financing was \(\$300\) less than the cost of taxes and license fees.

Answer: (D) \(\$300\) less

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "\(\frac{3}{5}\) of the remainder"
Students often misread this as "\(\frac{3}{5}\) of the total \(\$18,000\)" instead of understanding that it means \(\frac{3}{5}\) of what's left after subtracting fuel and maintenance costs. This leads to calculating depreciation as \(\frac{3}{5} \times \$18,000 = \$10,800\) instead of the correct \(\frac{3}{5} \times \$12,000 = \$7,200\).

2. Confusion with the insurance-financing relationship
The phrase "insurance was 3 times the cost of financing" can be misinterpreted. Some students might set up the equation as Financing = \(3 \times\) Insurance instead of Insurance = \(3 \times\) Financing, leading them to solve for the wrong variable first.

3. Overlooking that taxes/license fees are the remaining expenses
Students may not realize that taxes and license fees account for whatever money is left over after all other expenses are calculated. They might try to find a direct fractional relationship for taxes/license fees from the problem statement.

Errors while executing the approach

1. Arithmetic errors in fraction calculations
When calculating \(\frac{1}{3}\) of \(\$18,000\) or \(\frac{3}{5}\) of \(\$12,000\), students may make basic multiplication or division errors. For example, calculating \(\frac{3}{5} \times \$12,000\) as \(\$7,000\) instead of \(\$7,200\), or \(\frac{1}{5} \times \$18,000\) as \(\$3,000\) instead of \(\$3,600\).

2. Incorrect algebraic manipulation
When solving the system Insurance = \(3 \times\) Financing and Insurance + Financing = \(\$3,600\), students might incorrectly substitute or combine the equations, leading to wrong values for insurance and financing costs.

3. Addition errors when tracking total expenses
With multiple expense categories to track (\(\$6,000 + \$7,200 + \$2,700 + \$900\)), students may make addition errors when calculating the running total, leading to an incorrect amount for taxes/license fees.

Errors while selecting the answer

1. Reversing the comparison direction
After calculating that financing costs \(\$900\) and taxes/license fees cost \(\$1,200\), students might incorrectly state that financing was "\(\$300\) more" instead of "\(\$300\) less" than taxes/license fees, mixing up which value is larger.

2. Using absolute difference without direction
Students might correctly calculate the \(\$300\) difference but fail to determine whether financing was more or less than taxes/license fees, potentially guessing between "\(\$300\) more" and "\(\$300\) less."