Find the compound interest earned on a four year investment of $3,500 at 4.5% compounded monthly.

To find the compound interest earned, we use the compound interest formula:

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

where:
- $ A $ is the amount of money accumulated after n years, including interest.
- $ P $ is the principal amount (the initial sum of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for, in years.

We need to find the interest earned, which is $ A - P $.

Given:
- $ P = 3500 $
- $ r = 4.5\% = 0.045 $
- $ n = 12 $ (compounded monthly)
- $ t = 4 $ years

Substitute the given values into the formula:

\[ A = 3500 \left(1 + \frac{0.045}{12}\right)^{12 \times 4} \]

First, calculate $ \frac{0.045}{12} $:

\[ \frac{0.045}{12} = 0.00375 \]

Next, calculate $ 12 \times 4 $:

\[ 12 \times 4 = 48 \]

Now, calculate $ \left(1 + 0.00375\right)^{48} $:

\[ 1 + 0.00375 = 1.00375 \]

\[ (1.00375)^{48} \approx 1.1967 \]

Finally, calculate $ A $:

\[ A = 3500 \times 1.1967 \]

\[ A \approx 4188.45 \]

The compound interest earned is:

\[ \text{Compound Interest} = A - P \]

\[ \text{Compound Interest} = 4188.45 - 3500 \]

\[ \text{Compound Interest} \approx 688.45 \]

Therefore, the compound interest earned on a four-year investment of $3,500 at 4.5% compounded monthly is approximately $688.45.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. How does the frequency of compounding affect the amount of interest earned?
2. Can you explain the difference between nominal and effective interest rates?
3. How do you convert an annual interest rate to a monthly or daily rate?
4. What is continuous compounding, and how is it calculated?
5. How can I calculate the future value of an investment with compound interest?

**Tip:** When dealing with compound interest, the more frequently the interest is compounded, the higher the amount of interest earned over the same period.

Learn how to calculate compound interest on a $3,500 investment at 4.5% annual interest compounded monthly over four years. This detailed solution uses the compound interest formula and covers exponential growth concepts. Ideal for advanced high school students or anyone interested in financial calculations.

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