Calculate the length of time it will take each investment to double base on the given interest rate and the number of compoundings. Note: there will not be a lot of variation between your answers. So calculate carefully by rounding only your final calculation. Round to two decimal places as necessary.

Calculate Doubling Times (in years) when r=9 %



A)	Estimate using

the "Rule of 72"	

B)	Compounded Annually	

C)	Compounded Quarterly	

D)	Compounded Monthly	

E)	Compounded Continuously	

Question HelpQuestion 16: Message Message instructor

To calculate the length of time it takes for an investment to double at an interest rate of 9%, we'll use different methods: the Rule of 72, compounded annually, quarterly, monthly, and continuously. Here are the steps for each method:

### A) Estimate using the "Rule of 72"
The Rule of 72 is a simple way to estimate the doubling time of an investment. It states that the doubling time \( T \) (in years) can be approximated by dividing 72 by the annual interest rate \( r \) (expressed as a percentage):

\[
T \approx \frac{72}{r}
\]

For \( r = 9\% \):

\[
T \approx \frac{72}{9} = 8 \text{ years}
\]

### B) Compounded Annually
For annual compounding, the formula for the future value \( FV \) of an investment is:

\[
FV = PV \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( PV \) is the present value (initial investment),
- \( r \) is the annual interest rate (decimal),
- \( n \) is the number of times interest is compounded per year,
- \( t \) is the number of years.

We need to solve for \( t \) when \( FV = 2 \times PV \):

\[
2 = \left(1 + \frac{0.09}{1}\right)^{1 \cdot t}
\]

\[
2 = (1.09)^t
\]

Taking the natural logarithm of both sides:

\[
\ln(2) = t \ln(1.09)
\]

\[
t = \frac{\ln(2)}{\ln(1.09)} \approx \frac{0.693}{0.086} \approx 8.04 \text{ years}
\]

### C) Compounded Quarterly
For quarterly compounding (\( n = 4 \)):

\[
2 = \left(1 + \frac{0.09}{4}\right)^{4t}
\]

\[
2 = \left(1 + 0.0225\right)^{4t}
\]

\[
2 = (1.0225)^{4t}
\]

Taking the natural logarithm of both sides:

\[
\ln(2) = 4t \ln(1.0225)
\]

\[
t = \frac{\ln(2)}{4 \ln(1.0225)} \approx \frac{0.693}{4 \cdot 0.0222} \approx 7.91 \text{ years}
\]

### D) Compounded Monthly
For monthly compounding (\( n = 12 \)):

\[
2 = \left(1 + \frac{0.09}{12}\right)^{12t}
\]

\[
2 = \left(1 + 0.0075\right)^{12t}
\]

\[
2 = (1.0075)^{12t}
\]

Taking the natural logarithm of both sides:

\[
\ln(2) = 12t \ln(1.0075)
\]

\[
t = \frac{\ln(2)}{12 \ln(1.0075)} \approx \frac{0.693}{12 \cdot 0.00747} \approx 7.88 \text{ years}
\]

### E) Compounded Continuously
For continuous compounding, the formula is:

\[
FV = PV \cdot e^{rt}
\]

Solving for \( t \) when \( FV = 2 \times PV \):

\[
2 = e^{0.09t}
\]

Taking the natural logarithm of both sides:

\[
\ln(2) = 0.09t
\]

\[
t = \frac{\ln(2)}{0.09} \approx \frac{0.693}{0.09} \approx 7.70 \text{ years}
\]

### Summary of Doubling Times

- A) Rule of 72: **8.00 years**
- B) Compounded Annually: **8.04 years**
- C) Compounded Quarterly: **7.91 years**
- D) Compounded Monthly: **7.88 years**
- E) Compounded Continuously: **7.70 years**

Would you like more details on any of these calculations or have any questions?

---

**Further exploration questions:**

1. How does the Rule of 72 provide a quick estimation for doubling time?
2. Why does continuous compounding result in the shortest doubling time?
3. How does the frequency of compounding affect the time it takes for an investment to double?
4. Can you explain the difference between nominal and effective interest rates?
5. What happens to the doubling time if the interest rate is increased?
6. How would you calculate the doubling time for an interest rate compounded daily?
7. Why is the natural logarithm used in these calculations?
8. How does compound interest differ from simple interest in terms of growth over time?

**Tip:** Remember, the more frequently interest is compounded, the less time it takes for the investment to double.

Learn how to calculate the doubling times for investments at a 9% interest rate using various methods: Rule of 72, annual compounding, quarterly compounding, monthly compounding, and continuous compounding. Detailed step-by-step solutions provided with rounded final calculations.

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