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A certain insurance company pays verified claims on an individual basis as follows: \(0\%\) of the first \(\$500\) of the claim, \(80\%\) of the next \(\$2,500\) of the claim, and \(90\%\) of the amount of the claim in excess of \(\$3,000\). What is the total paid for two verified claims of \(\$2,000\) and \(\$25,000\), respectively?
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:
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
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:
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
For a $25,000 claim, this is large enough to hit all three tiers:
Let's break it down tier by tier:
Let's calculate each tier:
Total payment for $25,000 claim = $0 + $2,000 + $19,800 = $21,800
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 = \(\$1,200 + \$21,800 = \$23,000\)
The insurance company pays a total of $23,000 for both verified claims.
Looking at our answer choices, this matches option A) $23,000.
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.
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.
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.