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The problem involves evaluating a project with a main investment of $30,000, considering cash inflows over a period of 5 years. The task is to determine the following:

1. **Pay-Back Period and Profitability**
2. **Average Rate of Return (Before Tax)**

Here is how we can approach the solution:

### 1. **Pay-Back Period and Profitability**
The pay-back period is the time it takes for the cumulative cash inflows to recover the initial investment.

The cash inflows for the project are:

- Year 1: $10,000
- Year 2: $10,000
- Year 3: $15,000
- Year 4: $15,000
- Year 5: $25,000

Cumulative cash inflows:
- After Year 1: $10,000
- After Year 2: $20,000
- After Year 3: $35,000
- After Year 4: $50,000
- After Year 5: $75,000

The initial investment is $30,000. From the cumulative cash inflows, we can see that the initial investment is recovered sometime between Year 2 and Year 3.

- After Year 2: $20,000 (not fully recovered)
- After Year 3: $35,000 (investment recovered)

Thus, the **Pay-Back Period** is 2 years and a fraction. To find the exact time:
\[
\text{Time to recover in Year 3} = \frac{\$30,000 - \$20,000}{\$15,000} = \frac{\$10,000}{\$15,000} = \frac{2}{3} \text{ of a year} \approx 0.67 \text{ years}
\]
So, the exact Pay-Back Period = 2 years + 0.67 years ≈ **2.67 years**.

### 2. **Average Rate of Return (Before Tax)**

The Average Rate of Return (ARR) can be calculated using:
\[
\text{ARR} = \frac{\text{Average Annual Profit}}{\text{Initial Investment}} \times 100\%
\]

To compute ARR:
1. Calculate the total cash inflows over 5 years:
   \[
   \text{Total Cash Inflows} = 10,000 + 10,000 + 15,000 + 15,000 + 25,000 = 75,000
   \]
   
2. Average annual cash inflow (before tax and depreciation):
   \[
   \text{Average Annual Inflow} = \frac{75,000}{5} = 15,000
   \]
   
3. Depreciation is not mentioned, but assuming straight-line depreciation for $30,000 over 5 years:
   \[
   \text{Depreciation per year} = \frac{30,000}{5} = 6,000
   \]

4. Average annual profit before tax (assuming no other expenses except depreciation):
   \[
   \text{Average Annual Profit} = 15,000 - 6,000 = 9,000
   \]
   
5. Finally, ARR:
   \[
   \text{ARR} = \frac{9,000}{30,000} \times 100\% = 30\%
   \]

The **Average Rate of Return (Before Tax)** is **30%**.

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Would you like more details on any part of the solution or have any other questions?

Here are some related questions you might find helpful:
1. How do taxes impact the Average Rate of Return calculation?
2. What are the limitations of using the Pay-Back Period as an investment criterion?
3. How would the calculation change if there were additional operating expenses?
4. What if the inflows varied significantly each year—how would that affect the Pay-Back Period?
5. Can the ARR method be used for comparing different projects with different lifespans?

**Tip:** When considering investments, it's important to look at both the Pay-Back Period and the ARR, but also to consider other methods like Net Present Value (NPV) for a more comprehensive analysis.

Explore how to solve an investment analysis problem involving a $30,000 initial investment with cash inflows over 5 years. Learn to calculate the Pay-Back Period and Average Rate of Return (ARR) with detailed step-by-step solutions. Suitable for financial analysts and professionals.

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