https://file.aiquickdraw.com/mathbot/file/1729861812181h8ss2q5b.jpg

Let's break down the problems from the image and solve them step by step:

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### **Question 1:**
**What APR (to two places of decimal) is equivalent to 0.89% per month?**

To convert a monthly interest rate to an Annual Percentage Rate (APR), we use the formula:

\[
APR = (1 + r_{\text{monthly}})^{12} - 1
\]

Where \( r_{\text{monthly}} = 0.89\% = 0.0089 \).

Now calculate the APR:

\[
APR = (1 + 0.0089)^{12} - 1
\]
\[
APR = (1.0089)^{12} - 1 \approx 0.1126 = 11.26\%
\]

**Answer: 11.26%**

---

### **Question 2:**
**Calculate the Net Present Value (NPV) of Joe's investment using a discount rate of 7.3%.**

Joe's initial investment is €12,953, and the cash inflows for the next three years are:

- Year 1: €6,499
- Year 2: €4,239
- Year 3: €4,188

The NPV formula is:

\[
NPV = \sum \left( \frac{C_t}{(1 + r)^t} \right) - C_0
\]

Where:
- \( C_t \) is the cash inflow at time \( t \),
- \( r \) is the discount rate (7.3% = 0.073),
- \( C_0 \) is the initial investment (€12,953).

Now, calculating each discounted cash flow:

\[
NPV = \frac{6499}{(1 + 0.073)^1} + \frac{4239}{(1 + 0.073)^2} + \frac{4188}{(1 + 0.073)^3} - 12953
\]

\[
NPV = \frac{6499}{1.073} + \frac{4239}{1.151929} + \frac{4188}{1.23613} - 12953
\]

\[
NPV = 6057.96 + 3679.58 + 3387.68 - 12953
\]

\[
NPV \approx 13125.22 - 12953 = 172.22
\]

**Answer: €172.22**

---

### **Question 3:**
**Tom invests €1186 at 6.3% per year. How much is the investment worth after 6 years?**

We can use the compound interest formula:

\[
A = P \left(1 + \frac{r}{100}\right)^t
\]

Where:
- \( P = 1186 \) (initial investment),
- \( r = 6.3\% \) (annual interest rate),
- \( t = 6 \) years.

Substituting the values:

\[
A = 1186 \left(1 + \frac{6.3}{100}\right)^6
\]
\[
A = 1186 \times (1.063)^6
\]
\[
A \approx 1186 \times 1.448619 = 1717.96
\]

**Answer: €1717.96**

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Would you like further clarification on any of the questions? Here are five related questions that expand on the concepts:

1. How is APR calculated for different compounding periods, such as quarterly or daily?
2. How do changes in the discount rate affect the Net Present Value of an investment?
3. What is the difference between NPV and Internal Rate of Return (IRR)?
4. How is compound interest different from simple interest, and when should each be used?
5. How do taxes and inflation affect the future value of an investment?

**Tip:** Always remember to round to the appropriate decimal places, as financial problems often require precision based on the context (e.g., two decimal places for percentages).

Question 1: What APR (to two places of decimal), is equivalent to 0.89% per month? 

Question 2: Joe is considering investing in a machine, costing €12,953. The net cash inflows associated with the investment are; €6,499 in year 1, €4,239 in year 2, and €4,188 in year 3. Calculate the Net Present Value of this investment, using a discount rate of 7.3%. 

Question 3: Tom invests €1,186 at 6.3% per year. How much is the investment worth after 6 years?

Explore detailed solutions to finance-related questions, including converting a monthly interest rate to APR, calculating the Net Present Value (NPV) of an investment with a 7.3% discount rate, and finding the future value of an investment using compound interest. Suitable for Grades 10-12.

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