You deposit $100 each month into an account earning 6% compounded monthly.

Do not round numbers while working on the problem. Only round the final answer.

Since we are working with money, round to the cent.

﻿﻿﻿How much will you have in the account in 15 years?

﻿﻿﻿How much total money will you put into the account?

c) How much total interest will you earn?

To solve this problem, we need to calculate the future value of an annuity where monthly payments are made into an account that earns compound interest. We can use the **Future Value of an Annuity formula** for monthly compounding:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $ P $ is the monthly payment ($100).
- $ r $ is the monthly interest rate (6% annual rate divided by 12 months = 0.06 / 12 = 0.005).
- $ n $ is the total number of payments (15 years × 12 months/year = 180 months).

### a) Future Value of the Account

Let's plug in the values:

\[
FV = 100 \times \frac{(1 + 0.005)^{180} - 1}{0.005}
\]

First, calculate $ (1 + 0.005)^{180} $:

\[
(1.005)^{180} \approx 2.459600558
\]

Now, subtract 1 from this value:

\[
2.459600558 - 1 = 1.459600558
\]

Next, divide this result by the monthly interest rate $ r = 0.005 $:

\[
\frac{1.459600558}{0.005} \approx 291.9201116
\]

Finally, multiply this by the monthly payment $ P = 100 $:

\[
100 \times 291.9201116 \approx 29,192.01116
\]

So, the future value of the account after 15 years, rounded to the nearest cent, is **$29,192.01**.

### b) Total Money Deposited

The total money you will put into the account is simply the monthly payment multiplied by the number of payments:

\[
\text{Total Deposits} = 100 \times 180 = 18,000
\]

So, you will deposit **$18,000** into the account over 15 years.

### c) Total Interest Earned

The total interest earned is the difference between the future value and the total deposits:

\[
\text{Total Interest} = 29,192.01 - 18,000 = 11,192.01
\]

So, the total interest earned is **$11,192.01**.

Would you like more details or have any questions?

Here are some related questions:
1. What would be the future value if you increased the monthly deposit to $150?
2. How much interest would you earn if the account earned 7% instead of 6%?
3. What if the deposits were made quarterly instead of monthly?
4. How would early withdrawals affect the future value?
5. How does compounding frequency impact the total interest earned?

**Tip:** Consistent, smaller contributions can grow significantly over time due to the power of compound interest.

Learn how to calculate the future value of monthly deposits earning compound interest at 6% annually. Understand the total deposits made, total interest earned, and more. Suitable for advanced high school students and anyone interested in financial mathematics.

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