https://file.aiquickdraw.com/mathbot/file/17219316557798t7uvouk.png

Please give me the correct answer with accurate calculation.

You must round to the nearest cent as​ needed!

To solve this problem, we need to determine the future value of an annuity due, the total amount invested, and the interest earned. Given:

- Annual payment ($PMT$) = $11,000
- Number of years ($n$) = 3
- Annual interest rate ($r$) = 5%

An annuity due means that each payment is made at the beginning of each period. The future value ($FV$) of an annuity due is calculated using the formula:

\[ FV_{\text{annuity due}} = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \]

Where:
- $PMT$ is the annual payment.
- $r$ is the annual interest rate (as a decimal).
- $n$ is the number of years.

Let's perform the calculation step by step:

1. **Convert the interest rate to a decimal:**
\[ r = 5\% = 0.05 \]

2. **Calculate the future value of the annuity due:**
\[ FV_{\text{annuity due}} = 11,000 \times \left( \frac{(1 + 0.05)^3 - 1}{0.05} \right) \times (1 + 0.05) \]

First, calculate $(1 + r)^n$:
\[ (1 + 0.05)^3 = 1.157625 \]

Next, calculate $(1 + 0.05)^n - 1$:
\[ 1.157625 - 1 = 0.157625 \]

Then, divide by $r$:
\[ \frac{0.157625}{0.05} = 3.1525 \]

Finally, multiply by $PMT$ and $1 + r$:
\[ FV_{\text{annuity due}} = 11,000 \times 3.1525 \times 1.05 = 36,479.1375 \]

Rounding to the nearest cent:
\[ FV_{\text{annuity due}} \approx 36,479.14 \]

3. **Calculate the total amount invested:**
Since the payments are made at the beginning of each year for 3 years, the total amount invested is simply the sum of the payments:
\[ \text{Total amount invested} = 11,000 \times 3 = 33,000 \]

4. **Calculate the interest earned:**
\[ \text{Interest earned} = \text{Future Value} - \text{Total Amount Invested} \]
\[ \text{Interest earned} = 36,479.14 - 33,000 = 3,479.14 \]

So the answers are:
- Future value of the annuity: $\$36,479.14$
- Total amount invested: $\$33,000$
- Interest earned: $\$3,479.14$

Do you want more details or have any questions?

Here are 5 questions you can ask next:
1. How is the future value of an ordinary annuity calculated?
2. What is the difference between simple interest and compound interest?
3. How do you calculate the present value of an annuity due?
4. What is the formula for calculating the effective annual rate?
5. How do you determine the interest rate if you know the future value, present value, and number of periods?

**Tip:** When dealing with annuities, always verify if it is an ordinary annuity or an annuity due, as it affects the future value calculation.

Learn how to calculate the future value of an annuity due with $11,000 annual payments at 5% interest over 3 years. Understand the formula and steps involved in determining the total amount invested and interest earned. Suitable for advanced learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation