Under a certain company's medical insurance plan, the amount an employee must pay for medical expenses consists of payment for...

1. Translate the problem requirements: Understand the insurance payment structure - employee pays first $450 plus 20% of additional expenses, capped at $1,350 total. We need to find the minimum amount the plan could pay when employee paid the maximum.
2. Determine total medical expenses when employee pays maximum: Work backwards from the $1,350 maximum employee payment to find what total medical expenses would result in this payment.
3. Calculate the plan's payment: Subtract the employee's maximum payment from the total medical expenses to find what the plan paid.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down this insurance plan in plain English:

How the employee payment works:

• The employee always pays the first $450 of medical expenses
• For any expenses beyond $450, the employee pays 20% of those additional expenses
• BUT the employee never pays more than $1,350 total per year
• Everything else gets paid by the insurance plan

What we need to find:
If an employee paid exactly $1,350 (the maximum), what's the smallest amount the insurance plan could have paid?

Process Skill: TRANSLATE - Converting the insurance language into clear mathematical understanding

2. Determine total medical expenses when employee pays maximum

Let's work backwards from the $1,350 maximum payment.

Breaking down the $1,350 payment:

• First $450 goes toward the initial expenses (employee always pays this)
• The remaining amount: \(\$1,350 - \$450 = \$900\)

Since the employee pays 20% of additional expenses beyond the first $450, we can figure out how much those "additional expenses" were:

If $900 represents 20% of additional expenses, then:

• Additional expenses = \(\$900 ÷ 0.20 = \$4,500\)

Total medical expenses:

• First $450 + Additional $4,500 = $4,950

Let's verify: Employee pays \(\$450 + (20\% × \$4,500) = \$450 + \$900 = \$1,350\) ✓

Process Skill: INFER - Working backwards from the maximum payment to find total expenses

3. Calculate the plan's payment

Now we can find what the insurance plan paid:

Simple subtraction:

• Total medical expenses: $4,950
• Employee paid: $1,350
• Plan paid: \(\$4,950 - \$1,350 = \$3,600\)

Double-checking our work:

• Employee pays first $450: $450
• Employee pays 20% of remaining $4,500: \(0.20 × \$4,500 = \$900\)
• Total employee payment: \(\$450 + \$900 = \$1,350\) ✓
• Plan pays the rest: \(\$4,950 - \$1,350 = \$3,600\) ✓

4. Final Answer

The least amount that the plan could have paid is $3,600.

This matches answer choice C. $3,600.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the payment structure
Students often confuse the layered payment system. They might think the employee pays 20% of ALL medical expenses rather than understanding that the employee pays 100% of the first $450 PLUS 20% of any additional expenses beyond that initial $450. This fundamental misreading leads to completely incorrect calculations.

2. Misinterpreting "least amount the plan could have paid"
Some students get confused by this wording and think they need to find a scenario where the plan pays less money. They don't realize that when the employee pays the maximum $1,350, there's only ONE possible total expense amount, which means there's only ONE amount the plan could have paid - not multiple scenarios to choose from.

3. Focusing on the wrong constraint
Students might overlook that the employee paid "the maximum amount" ($1,350) and instead try to work with other constraints or create hypothetical scenarios. Missing this key constraint means they can't properly set up the problem to work backwards from the maximum payment.

Errors while executing the approach

1. Arithmetic errors in percentage calculations
When calculating that $900 represents 20% of additional expenses, students often make errors like: \(\$900 × 0.20 = \$180\) (instead of \(\$900 ÷ 0.20 = \$4,500\)). This confusion between multiplying by the percentage versus dividing by it is common when working backwards from a percentage.

2. Incorrect breakdown of the $1,350 payment
Students might incorrectly assume the entire $1,350 is subject to the 20% rule, forgetting that $450 of it represents the full payment for the first $450 of expenses. This leads them to calculate: \(\$1,350 ÷ 0.20 = \$6,750\) in additional expenses rather than \((\$1,350 - \$450) ÷ 0.20 = \$4,500\).

3. Adding amounts in wrong sequence
Even with correct individual calculations, students sometimes add the amounts incorrectly. For example, they might calculate the additional expenses as $4,500 but then forget to add back the initial $450, leading to a total medical expense of $4,500 instead of $4,950.

Errors while selecting the answer

1. Selecting an intermediate calculation as the final answer
Students often select $4,500 (answer choice D) because this represents the additional expenses beyond the first $450, or they might select $1,080 (answer choice B) through some miscalculation. They lose track of what the question is actually asking for - the amount the plan paid.

2. Confusing what the plan paid versus what the employee paid
Some students calculate everything correctly but then select $1,350 as their answer (which isn't even listed), thinking the question asks for the employee payment rather than the plan payment. Or they might accidentally select an amount that represents some portion of the employee's payment structure.