You are going to deposit $22,000 today. You will earn an annual rate of 4.5 percent for 13 years, and then earn an annual rate of 3.9 percent for 16 years. How much will you have in your account in 29 years?

To find out how much you will have in your account after 29 years, we need to apply compound interest in two stages, since the interest rate changes after 13 years.

### Step 1: Calculate the amount after 13 years at 4.5% annual interest

We use the compound interest formula:

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

where:
- $ A $ is the amount of money accumulated after interest,
- $ P $ is the principal amount (\$22,000),
- $ r $ is the annual interest rate (4.5% or 0.045),
- $ n $ is the number of times the interest is compounded per year (for simplicity, we assume it's compounded annually, so $ n = 1 $),
- $ t $ is the time the money is invested for (13 years in this step).

\[
A_{13} = 22000 \times \left(1 + \frac{0.045}{1}\right)^{13}
\]

### Step 2: Calculate the amount after the next 16 years at 3.9% annual interest

The amount from Step 1 becomes the new principal for the next 16 years. Now, we apply the same compound interest formula, but with a 3.9% annual interest rate:

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

where:
- $ P $ is now the amount $ A_{13} $ from Step 1,
- $ r = 0.039 $ (3.9% annual interest),
- $ t = 16 $ years.

\[
A_{29} = A_{13} \times \left(1 + \frac{0.039}{1}\right)^{16}
\]

Now, let me calculate this for you.After 29 years, your account will have approximately **$71,909**.

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

1. How would the final amount change if the interest was compounded quarterly instead of annually?
2. What happens if the interest rates were switched, earning 3.9% for the first 13 years and 4.5% for the last 16 years?
3. How much interest will you have earned in total after 29 years?
4. How long would it take for your money to double at a constant interest rate of 4.5%?
5. What would the final amount be if you added annual contributions of $1,000?

**Tip:** When dealing with changing interest rates, always break the calculation into segments to apply the correct rate for each period.

Discover how to calculate the future value of an investment using compound interest with changing rates. In this example, we deposit $22,000 at a 4.5% interest rate for 13 years, followed by 3.9% for 16 years. Perfect for high school students learning about compound interest and exponential growth.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation