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A woman invests $1,600 at a rate of 8.6%. Find the time in years that it takes her investment to double with annual compounding (a) using the future value formula and (b) using the rule of 72.

A) based on the future value formula, it will take approximately _ years for her investment to double. (Round to two decimal places as needed.)

B) Based on the rule of 72, it will take approximately _ years for her investment to double. (Round to two decimal places as needed.)

For the first problem, please use: A=P (1+ r/m) ^n

For the second problem you would do years to double = blank/growth rate

Need Help With This Problem A Woman Invests 1600 At A Rate Of 86 Find The Time In Years That It Takes Her Investment To Double With Annual Compounding A Using T class=

Sagot :

Answer:

Step-by-step explanation: Let's solve the problem step-by-step for both parts using the given methods:

**A) Using the Future Value Formula:**

The future value formula for compound interest is:

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

where:

- \( A \) is the future value of the investment,

- \( P \) is the principal amount (initial investment),

- \( r \) is the annual interest rate (as a decimal),

- \( m \) is the number of compounding periods per year,

- \( t \) is the number of years.

In this case:

- \( P = 1600 \) dollars,

- \( r = 8.6\% = 0.086 \) (decimal form),

- We want \( A = 3200 \) dollars (double the initial investment),

- Compounded annually (\( m = 1 \)).

Let's solve for \( t \):

\[ 3200 = 1600 \left(1 + \frac{0.086}{1}\right)^{1 \cdot t} \]

Divide both sides by 1600:

\[ 2 = \left(1 + 0.086\right)^t \]

Take the natural logarithm (ln) of both sides to solve for \( t \):

\[ \ln(2) = t \cdot \ln(1.086) \]

Now, solve for \( t \):

\[ t = \frac{\ln(2)}{\ln(1.086)} \]

Using a calculator:

\[ t \approx \frac{0.693147}{0.083381} \approx 8.32 \]

So, using the future value formula, it will take approximately **8.32 years** for her investment to double.

**B) Using the Rule of 72:**

The Rule of 72 states that you can estimate the number of years it takes for an investment to double by dividing 72 by the annual interest rate (as a percentage):

\[ \text{Years to double} = \frac{72}{\text{annual interest rate}} \]

In this case:

\[ \text{Years to double} = \frac{72}{8.6} \approx 8.37 \]

Rounded to two decimal places, using the Rule of 72, it will take approximately **8.37 years** for her investment to double.

### Summary:

- **A)** Using the future value formula: Approximately **8.32 years**.

- **B)** Using the Rule of 72: Approximately **8.37 years**.

These calculations provide two different methods to estimate the time it takes for an investment to double under annual compounding with an interest rate of 8.6%.

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