Len wants to save money to purchase a car. She buys an annuity with monthly payments that earn 2% intertest, compounded monthly. Payments will be made at the end of each month. find the total value of the annuity in 5 years if each monthly payment is $279. 

To find the total value of an annuity with regular monthly payments and interest compounded monthly, we use the future value formula for an ordinary annuity. In this case, Len makes monthly payments at the end of each month, which qualifies as an ordinary annuity.

### Future Value of an Ordinary Annuity Formula

The future value $FV$ of an ordinary annuity is given by:

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

where:
- $ P $ is the payment amount per period,
- $ r $ is the monthly interest rate (in decimal form),
- $ n $ is the total number of payments.

### Given Values

- Monthly payment, $ P = 279 $ dollars,
- Annual interest rate, $ 2\% $, compounded monthly,
- Number of years, $ 5 $ years.

### Step-by-Step Calculation

1. **Convert the annual interest rate to a monthly interest rate:**

\[
r = \frac{2\%}{12} = \frac{0.02}{12} \approx 0.001667
\]

2. **Determine the total number of payments:**

\[
n = 5 \times 12 = 60
\]

3. **Calculate the future value $ FV $ using the formula:**

\[
FV = 279 \times \frac{(1 + 0.001667)^{60} - 1}{0.001667}
\]

Let's compute the future value step by step:

\[
1 + 0.001667 = 1.001667
\]

\[
(1.001667)^{60} \approx 1.10304
\]

\[
1.10304 - 1 = 0.10304
\]

\[
\frac{0.10304}{0.001667} \approx 61.82
\]

\[
FV = 279 \times 61.82 \approx 17247.78
\]

### Conclusion

The total value of the annuity after 5 years is approximately **$17,247.78**.

### Do you want more details or have any questions?

Here are some related questions to consider:

1. What would be the future value of the annuity if the interest rate was 3% instead of 2%?
2. How much interest does Len earn on her annuity over the 5 years?
3. How does changing the compounding frequency affect the future value of the annuity?
4. What would be the future value if Len decides to make payments at the beginning of each month instead (an annuity due)?
5. How many months would it take for Len's annuity to reach a future value of $20,000 with the same payment and interest rate?

**Tip:** Remember to convert annual rates to the appropriate period rates when working with financial formulas involving compounding.

Learn how to calculate the future value of an annuity with monthly payments earning 2% interest compounded monthly. Use our step-by-step guide with detailed calculations. Discover the total value after 5 years with monthly payments of $279.

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