what would the interest earned be if the interest rate was compounded annually instead of being simple?

To find the interest earned with annual compounding, we'll use the formula for compound interest:

\[
A = P \times \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount ($39,844).
- $ r $ is the annual interest rate (we found it to be 2.075% or 0.02075).
- $ n $ is the number of times interest is compounded per year (since it's compounded annually, $ n = 1 $).
- $ t $ is the time the money is invested for (10 years).

The interest earned is then $ A - P $.

First, let's calculate the total amount $ A $:

\[
A = 39844 \times \left(1 + \frac{0.02075}{1}\right)^{1 \times 10}
\]

Now, let's perform the calculation:

\[
A = 39844 \times \left(1.02075\right)^{10}
\]

\[
A \approx 39844 \times 1.22863 \approx 48981.99
\]

Now, subtract the principal to find the interest earned:

\[
\text{Interest Earned} = A - P = 48981.99 - 39844 \approx 9137.99
\]

So, the interest earned with annual compounding would be approximately **$9,138**.

Would you like more details or have any questions?

---

Here are some related questions you might find useful:

1. How does compounding frequency (monthly, quarterly) affect the interest earned?
2. What would the interest earned be if the rate was 3% with the same compounding method?
3. How does inflation affect the real value of the interest earned over time?
4. Can you explain the concept of the "effective annual rate" (EAR)?
5. How would the interest calculation change if the investment period was extended to 15 years?
6. What happens if interest rates change periodically over the 10 years?
7. How would the calculation change if the interest rate were compounded semi-annually?
8. What is the relationship between the compound interest formula and the exponential function?

**Tip:** Compounding can significantly increase your returns over time. The more frequently the interest is compounded, the higher the total amount you'll accumulate.

Learn how to calculate interest earned with annual compounding using the compound interest formula. In this example, find out the interest earned on a principal amount of $39,844 with an annual interest rate of 2.075% over 10 years. Get a step-by-step solution and understand how compounding affects your investment.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation