A certain insurance company pays verified claims on an individual basis as follows: 0% of the first $500 of the...

1. Translate the problem requirements: The insurance company uses a tiered payment system: 0% of first $500, 80% of next $2,500 (amounts from $501-$3,000), and 90% of any amount above $3,000. We need to calculate total payments for claims of $2,000 and $25,000.
2. Apply payment tiers to the $2,000 claim: Break down this claim into the applicable tiers and calculate payment for each tier, then sum them up.
3. Apply payment tiers to the $25,000 claim: Break down this larger claim across all three tiers and calculate the payment for each tier, then sum them up.
4. Sum both claim payments: Add the total payments from both claims to get the final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this insurance company actually does in plain English:

Think of it like a three-tier system where different parts of your claim get different treatment:

• Tier 1 (First $500): You get nothing back - 0% coverage
• Tier 2 (Next $2,500, so from $501 to $3,000): You get 80% back
• Tier 3 (Everything above $3,000): You get 90% back

It's like a progressive tax system, but in reverse - the higher your claim, the better your coverage gets.

We need to calculate how much the insurance company pays for two separate claims: one for $2,000 and another for $25,000.

Process Skill: TRANSLATE - Converting the tiered payment structure into clear, actionable brackets

2. Apply payment tiers to the $2,000 claim

For a $2,000 claim, let's see which tiers this falls into:

Since $2,000 is less than $3,000, this claim only touches the first two tiers:

• Tier 1: First $500 → Company pays 0% of $500 = $0
• Tier 2: Remaining amount = $2,000 - $500 = $1,500 → Company pays 80% of $1,500

Let's calculate the Tier 2 payment:

80% of $1,500 = \(0.80 \times \$1,500 = \$1,200\)

Total payment for $2,000 claim = $0 + $1,200 = $1,200

3. Apply payment tiers to the $25,000 claim

For a $25,000 claim, this is large enough to hit all three tiers:

Let's break it down tier by tier:

• Tier 1: First $500 → Company pays 0% of $500 = $0
• Tier 2: Next $2,500 (from $501 to $3,000) → Company pays 80% of $2,500
• Tier 3: Remaining amount = $25,000 - $3,000 = $22,000 → Company pays 90% of $22,000

Let's calculate each tier:

• Tier 1 payment: $0
• Tier 2 payment: 80% of $2,500 = \(0.80 \times \$2,500 = \$2,000\)
• Tier 3 payment: 90% of $22,000 = \(0.90 \times \$22,000 = \$19,800\)

Total payment for $25,000 claim = $0 + $2,000 + $19,800 = $21,800

4. Sum both claim payments

Now we simply add up what the insurance company pays for both claims:

Payment for $2,000 claim: $1,200

Payment for $25,000 claim: $21,800

Total payment for both claims = \(\$1,200 + \$21,800 = \$23,000\)

Final Answer

The insurance company pays a total of $23,000 for both verified claims.

Looking at our answer choices, this matches option A) $23,000.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the tier boundaries: Students often misunderstand that the $2,500 in "next $2,500" refers to amounts from $501 to $3,000, not from $0 to $2,500. This leads them to incorrectly apply the 80% rate to the wrong portion of the claim.

2. Treating it as a simple percentage problem: Students may try to apply a single percentage rate to the entire claim amount instead of recognizing this as a tiered system where different portions get different treatment, similar to tax brackets.

3. Confusion about cumulative vs. incremental coverage: Students might think that if a claim exceeds $3,000, the entire amount gets 90% coverage, rather than understanding that only the portion above $3,000 gets the highest rate.

Errors while executing the approach

1. Arithmetic errors with percentage calculations: Simple computational mistakes like calculating 80% of $2,500 as $2,000 instead of $2,000, or 90% of $22,000 as $19,800 instead of the correct amount can lead to wrong final answers.

2. Incorrect tier amount calculations: For the $25,000 claim, students might calculate the Tier 3 amount as $25,000 - $2,500 = $22,500 instead of the correct $25,000 - $3,000 = $22,000, forgetting that Tier 2 goes up to $3,000 total.

3. Missing or double-counting tier amounts: Students may forget to include one of the tiers in their final calculation or accidentally count a tier twice, especially when working with the larger $25,000 claim that spans all three tiers.

Errors while selecting the answer

1. Adding only one claim payment: Students might correctly calculate both individual claim payments ($1,200 and $21,800) but then select an answer choice that represents only one of these amounts instead of their sum of $23,000.

2. Confusing total claims vs. total payments: Students might mistakenly select $25,000 (choice E) thinking the question asks for the total claim amount rather than the total amount the insurance company actually pays out.