Double your wealth. Kant Miss Company is promising its investors that it will double their money every 5 years. What annual rate is Kant Miss promising? Is this investment a good deal? If you invest $$ \$ 450 $$ now and Kant Miss is able to deliver on its promise, how long will it take your investment to reach $$ \$ 35,000 $$ ? Using the Rule of 72 , what annual rate is Kant Miss promising? $$ 14 \% $$ (Round to the nearest whole percentage.) Using the time value of money equation, what annual rate is Kant Miss promising? $$ \square \% $$ (Round to two decimal places.)

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To determine the exact annual rate Kant Miss Company is promising to double your investment in 5 years, we can use the **Time Value of Money (TVM) equation**:

$$
FV = PV \times (1 + r)^n
$$

Where:
- $$ FV $$ = Future Value
- $$ PV $$ = Present Value
- $$ r $$ = Annual interest rate
- $$ n $$ = Number of years

Given that the investment doubles, $$ FV = 2 \times PV $$. Plugging the values into the equation:

$$
2 = (1 + r)^5
$$

To solve for $$ r $$:

1. **Take the natural logarithm of both sides:**

$$
\ln(2) = 5 \times \ln(1 + r)
$$

2. **Isolate $$ \ln(1 + r) $$:**

$$
\ln(1 + r) = \frac{\ln(2)}{5} \approx \frac{0.69314718056}{5} \approx 0.1386294361
$$

3. **Exponentiate both sides to solve for $$ 1 + r $$:**

$$
1 + r = e^{0.1386294361} \approx 1.148698355
$$

4. **Subtract 1 to find $$ r $$:**

$$
r \approx 0.148698355 \text{ or } 14.87\%
$$

**Answer:** Kant Miss Company is promising an annual rate of **14.87%**.

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### **Is This Investment a Good Deal?**

An annual return of **14.87%** is significantly higher than the average returns from traditional investments like savings accounts, bonds, or even the stock market. However, higher returns typically come with higher risk. It's essential to assess the credibility of Kant Miss Company, understand the nature of the investment, and evaluate the associated risks before committing your funds.

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### **Time to Grow \$450 to \$35,000**

Assuming Kant Miss delivers a consistent annual rate of **14.87%**, we can calculate the time it takes for \$450 to grow to \$35,000 using the TVM equation:

$$
FV = PV \times (1 + r)^n
$$

$$
35,000 = 450 \times (1 + 0.1487)^n
$$

$$
\frac{35,000}{450} \approx 77.7778 = (1.1487)^n
$$

Taking natural logarithm on both sides:

$$
\ln(77.7778) = n \times \ln(1.1487)
$$

$$
n = \frac{\ln(77.7778)}{\ln(1.1487)} \approx \frac{4.3553}{0.1386} \approx 31.43 \text{ years}
$$

**It would take approximately 31.43 years for your \$450 investment to reach \$35,000 at an annual rate of 14.87%.**

---

**Summary of Answers:**
- **Using the Rule of 72:** Approximately **14%**
- **Using the Time Value of Money Equation:** **14.87%** (rounded to two decimal places)
Kant Miss Company is promising an annual rate of **14.87%**. Using the Rule of 72, this is approximately **14%**.

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