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The problem provided deals with calculating how much a person would pay with and without an insurance policy that includes a deductible. Let’s break down and solve part (a) of the question based on the information:

### Information Provided:
- The person has an insurance policy with an **$800 deductible per claim** for collisions.
- There are two claims over a two-year period: one for **$540** and one for **$1000**.
- The **annual premium** for the policy is **$550**.
  
### (a) Calculations:

#### 1. **With Insurance Policy**:
   - **Claim 1**: $540 claim (below the $800 deductible):
     - Since the deductible is higher than the claim, the person will pay the full amount of **$540**.
   
   - **Claim 2**: $1000 claim (above the $800 deductible):
     - The person will pay the deductible of **$800**.
     - The insurance will cover the remaining $1000 - $800 = **$200**.

   - **Total out-of-pocket cost for claims**: 
     - $540 (for the first claim) + $800 (deductible for the second claim) = **$1340**.
   
   - **Add the annual premium** for the insurance policy over two years:
     - $550/year × 2 = **$1100**.

   - **Total cost with insurance**: 
     - $1340 (for claims) + $1100 (for premiums) = **$2440**.

#### 2. **Without Insurance Policy**:
   - Without insurance, the person will pay the full amount for both claims.
   - Total cost: $540 (for the first claim) + $1000 (for the second claim) = **$1540**.

### Summary:
- **With insurance, the person would pay**: $2440.
- **Without insurance, the person would pay**: $1540.

Would you like further clarification or help with parts (b) through (d)?

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Here are five related questions to consider:
1. How would the cost change if the deductible was different?
2. What if the person had additional claims during the year?
3. How does the annual premium affect the overall savings with insurance?
4. How might a higher or lower deductible influence the decision to have insurance?
5. How does this model change if the insurance covers more than just collisions?

**Tip**: Always compare the total costs with and without insurance over time to see if the premiums are worth it.

A person has a car insurance policy with a $800 deductible provision (per claim) for collisions. During a two-year period, the person files claims for $540 and $1000. The annual premium for the policy is $550. With the insurance policy, the person would pay ____. Without the insurance policy, the person would pay ____.

Learn how to calculate the total costs a person would pay with and without an insurance policy. This example involves a car insurance policy with an $800 deductible and covers claims of $540 and $1000 over two years, with an annual premium of $550. Understand the difference in costs using simple arithmetic calculations.

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