Professor Smith puts $100,000 in a retirement account for 20 years. The account gives an annual interest rate of 4.15%, compounded quarterly. (a) How much is in the account at the end of 20 years? (b) How much interest did he earn on his investment during the 20 years?

To solve this, we use the compound interest formula:

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

Where:
- $A$ is the amount of money in the account after interest.
- $P$ is the initial principal (the amount of money initially invested).
- $r$ is the annual interest rate (decimal).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the time the money is invested (in years).

Given:
- $P = 100,000$ dollars
- $r = 4.15\%$ or 0.0415
- $n = 4$ (quarterly compounding)
- $t = 20$ years

### (a) Calculate the amount $A$ in the account at the end of 20 years:
Substitute the values into the compound interest formula:

\[
A = 100,000 \left(1 + \frac{0.0415}{4}\right)^{4 \times 20}
\]

First, calculate $ \frac{0.0415}{4} $:

\[
\frac{0.0415}{4} = 0.010375
\]

Now calculate the exponent $4 \times 20 = 80$, so the formula becomes:

\[
A = 100,000 \left(1 + 0.010375\right)^{80}
\]

\[
A = 100,000 \left(1.010375\right)^{80}
\]

Now calculate $ \left(1.010375\right)^{80} $:

\[
A \approx 100,000 \times 2.25957
\]

\[
A \approx 225,957
\]

So, the amount in the account at the end of 20 years is approximately **$225,957**.

### (b) Calculate the interest earned:
The interest earned is the total amount in the account minus the initial principal:

\[
\text{Interest} = A - P = 225,957 - 100,000 = 125,957
\]

So, the interest earned over 20 years is approximately **$125,957**.

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Do you want further details on the steps? Here are five related questions:

1. What would be the total amount if the interest rate were 5% instead of 4.15%?
2. How would the result change if the interest were compounded annually instead of quarterly?
3. What is the formula for continuous compounding interest, and how does it differ?
4. How much interest is earned after 10 years with the same conditions?
5. What would be the future value of monthly compounded interest with the same rate and time period?

**Tip:** When comparing different compounding periods, more frequent compounding (e.g., quarterly vs. annually) usually leads to higher returns.

Learn how to calculate compound interest for a $100,000 investment over 20 years with a 4.15% interest rate, compounded quarterly. This step-by-step solution explains the key formula for compound interest and provides insights into exponential growth and interest rates. Suitable for students in Grades 10-12.

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