An initial investment amount​ P, an annual interest rate​ r, and a time t are given. Find the future value of the investment when interest is compounded​ (a) annually,​ (b) monthly,​ (c) daily, and​ (d) continuously. Then find​ (e) the doubling time T for the given interest rate.
Pequals​$1500​, requals3.15​%, tequals5 yr
Question content area bottom
Part 1
​a) The future value of the investment when interest is compounded annually is ​$

enter your response here.
​(Type an integer or a decimal. Round to the nearest cent as​ needed.)

To find the future value of an investment $ P $ with an annual interest rate $ r $ over a time period $ t $ when interest is compounded in different ways, we use the following formulas:

### (a) Compounded Annually

The formula for future value when interest is compounded annually is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- $ P $ is the principal amount (\$1500),
- $ r $ is the annual interest rate (3.15% or 0.0315 as a decimal),
- $ n $ is the number of times interest is compounded per year (annually, $ n = 1 $),
- $ t $ is the time in years (5 years).

Substituting the given values:
\[ A = 1500 \left(1 + \frac{0.0315}{1}\right)^{1 \times 5} \]
\[ A = 1500 \left(1 + 0.0315\right)^5 \]
\[ A = 1500 \left(1.0315\right)^5 \]
\[ A \approx 1500 \times 1.168811 \]
\[ A \approx 1753.22 \]

So, the future value of the investment when compounded annually is approximately **\$1753.22**.

Would you like details or have any questions? Here are some related questions to expand on the topic:

1. How does the future value change if the interest rate increases?
2. What is the formula for continuous compounding, and why is it different?
3. How does the compounding frequency affect the future value of an investment?
4. What is the future value when compounded monthly?
5. What is the future value when compounded daily?
6. How is the doubling time $ T $ calculated for a given interest rate?
7. What is the impact of extending the investment period to 10 years?
8. How does decreasing the principal amount affect the future value?

**Tip:** Always round your final answers to the nearest cent when dealing with currency to maintain accuracy in financial calculations.

Learn how to calculate the future value of an investment compounded annually, monthly, daily, and continuously. Use the compound interest formula and understand the concept of doubling time. Example calculation provided with an initial investment amount of $1500, an annual interest rate of 3.15%, and a time period of 5 years.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation