Problem 14 - Thinking Mathematically

Question

A service that repairs air conditioners sells maintenance agreements for \(\$ 80\) a year. The average cost for repairing an air conditioner is \(\$ 350\) and 1 in every 100 people who purchase maintenance agreements have air conditioners that require repair. Find the service's expected profit per maintenance agreement.

Step-by-Step Solution

Answer

The service's expected profit per maintenance agreement is approximately $76.50.

Step 1: Identify the Revenue

The selling price for a maintenance agreement is $80, this constitutes the revenue per agreement sold.

Step 2: Calculate the Expected Repair Cost

The average repair cost is $350 and, according to the problem, this expense will only be incurred for 1 in every 100 customers. Therefore, the expected cost per maintenance agreement sold is \(\frac{350}{100} = \$3.50\)

Step 3: Compute the Expected Profit

The profit per maintenance agreement can be found by subtracting the expected repair cost ($3.50) from the revenue per agreement ($80). The expected profit per agreement is therefore \(80 - 3.5 = \$76.50\)

Key Concepts

Understanding Probability in Maintenance Agreements

Probability plays a crucial role in financial decisions, especially when it comes to maintenance agreements. In this context, probability refers to the chance that a customer's air conditioner will require repair within a year. Here, the probability of needing a repair is 1 in 100.
This means that out of every 100 maintenance agreements sold, only one is expected to result in a repair.

• Calculation of Probability: The given probability of occurrence is 1/100, or 0.01. This is determined from historical data or statistical analysis.
• Application of Probability: This probability influences how the company plans its finances and estimates its expected costs and profits.

Understanding probability helps the company predict future repair occurrences and balance their financial strategies accordingly.

Profit Calculation for Maintenance Agreements

Calculating expected profit involves analyzing all potential revenues and costs linked with selling maintenance agreements. The revenue is what the company earns per agreement sold, while the cost is primarily the repair expenses.
The expected profit can be calculated by taking into account not only the straightforward revenues and costs but also the probabilities of those costs being incurred.

• Revenue Determination: For each agreement sold, the company gains $80, which serves as its revenue.
• Expected Repair Cost: Given a probability of 1 in 100, the expected cost per agreement for potential repairs is calculated as \(\frac{350}{100} = 3.50\).
• Calculation of Expected Profit: By subtracting the expected repair costs from the revenue, the profit per agreement is found: \(80 - 3.50 = 76.50\).

These calculations provide the company with a clear view of potential gains and help in making informed business decisions.

Importance of Maintenance Agreements

Maintenance agreements are essential for both businesses and customers, as they provide a systematic approach to handling repair services.
Such agreements assure customers of hassle-free repairs, while companies secure a steady revenue stream.

Key Benefits of Maintenance Agreements:

• For Customers: Peace of mind knowing that potential large expenses, like costly repairs, are covered.
• For Companies: Generates upfront revenue and minimizes repair costs due to calculated risk management.

These agreements also improve the company's customer service reputation by showing commitment to taking care of client needs without unexpected fees. In summary, maintenance agreements benefit both parties through risk management and financial predictability.