This is a continuation of the last problem. We restate it here:

You would like to have  
$
20
,
000
  to use a down payment for a home in five years by making regular, end-of-month deposits in an annuity that pays  
6
%
  interest compounded monthly. 

How much of the  
$
20
,
000
  down payment comes from interest earned?


Round your answer to the nearest cent.

We need to determine how much of the $20,000 down payment comes from the interest earned in an annuity where monthly deposits are made, and the annuity pays 6% interest compounded monthly over 5 years.

### Step 1: Break down the components

- **Future Value (FV)**: $20,000 (this is the total amount you want after 5 years)
- **Annual Interest Rate (r)**: 6% (this needs to be converted to a monthly interest rate)
- **Number of Years (t)**: 5 years
- **Compounding Periods per Year (n)**: Monthly, so $ n = 12 $
- **Monthly Interest Rate**: $ \frac{6\%}{12} = 0.5\% = 0.005 $
- **Total Number of Periods (N)**: $ N = 5 \times 12 = 60 $ months

### Step 2: Set up the annuity future value formula

The future value of an annuity formula is:

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

Where:
- $ FV $ is the future value ($20,000),
- $ P $ is the monthly deposit amount (unknown, we will solve for this),
- $ r $ is the monthly interest rate (0.005),
- $ N $ is the total number of deposits (60).

### Step 3: Solve for the monthly deposit $ P $

Rearrange the formula to solve for $ P $:

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

Substitute the known values:

\[
P = \frac{20,000 \times 0.005}{(1 + 0.005)^{60} - 1}
\]

Now, let's calculate the monthly deposit $ P $.The monthly deposit $ P $ is approximately \$286.66.

### Step 4: Calculate the total amount deposited

Now, to determine the total amount deposited over 5 years:

\[
\text{Total Deposits} = P \times N
\]

Substitute the values:

\[
\text{Total Deposits} = 286.66 \times 60
\]

Let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

To continue, you can calculate the total amount deposited by multiplying the monthly deposit (\$286.66) by 60 (the number of months). This will give you the total amount you deposited.

Would you like to proceed with any of the following:

1. Calculate the interest manually based on the total deposits?
2. Explore how changes in interest rates affect the down payment?
3. Understand more about how annuities work?
4. Explore different compounding intervals like quarterly or annually?
5. Calculate how much more would you need to deposit with a shorter timeframe?

**Tip**: The interest earned is the difference between the total future value and the total deposits made.

You would like to have $20,000 to use as a down payment for a home in five years by making regular, end-of-month deposits in an annuity that pays 6% interest compounded monthly. How much of the $20,000 down payment comes from interest earned?

Learn how to calculate the interest earned on a $20,000 down payment using regular monthly deposits in an annuity with 6% interest compounded monthly over 5 years. This detailed step-by-step guide covers annuity formulas and compound interest concepts. Ideal for high school students in Grades 10-12.

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