You would like to have $5,000.00 in 15 years for a special vacation following graduation by making deposits at the end of each six months in an annuity that pays 3.8% compounded semiannually.

How much money should you deposit at the end of every six months?
Incorrect Do not round until the final answer, and then round up to the next penny.)

How much of the $5,000.00 comes from deposits and how much comes from interest?
$Incorrect of the $5,000.00 comes from your deposits and $

We can approach this problem using the formula for the future value of an ordinary annuity, which is used when periodic payments are made at the end of each period. The formula is:

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

Where:
- $ FV $ is the future value (the total amount you want, $5,000).
- $ P $ is the amount of each periodic deposit (the amount we're solving for).
- $ r $ is the interest rate per period.
- $ n $ is the total number of periods.

Given:
- $ FV = 5,000 $
- Annual interest rate = 3.8%, compounded semiannually, so the rate per period $ r = \frac{3.8\%}{2} = 0.019 = 0.019 $.
- Total time = 15 years, with 2 periods per year, so $ n = 15 \times 2 = 30 $ periods.

We'll solve for $ P $, the periodic deposit.

### Step 1: Solve for the periodic deposit $ P $
We rearrange the formula to solve for $ P $:

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

Substituting the known values:

\[
P = \frac{5,000}{\frac{(1 + 0.019)^{30} - 1}{0.019}}
\]

Let me calculate that for you.The amount you should deposit at the end of every six months is approximately **$125.19**.

### Step 2: Determine how much of the $5,000 comes from deposits and how much comes from interest.

Now, let's calculate the total amount deposited over 30 periods and subtract that from the future value to determine the amount contributed by interest.

- Total amount from deposits = $ P \times n $
- Interest earned = Future value - Total deposits

Let me calculate that for you.Out of the $5,000.00:

- **$3,755.75** comes from your deposits.
- **$1,244.25** comes from interest earned over the 15 years.

Would you like more details or have any questions? 

Here are 5 related questions:
1. How would the required deposit change if the interest rate increased to 4%?
2. What if the compounding frequency was quarterly instead of semiannual?
3. How long would it take to save $5,000 if you deposited $150 every six months?
4. What would happen to the interest earned if you doubled your deposits?
5. How does an ordinary annuity differ from an annuity due?

**Tip:** Always verify whether compounding periods match your deposit intervals for accurate calculations.

You would like to have $5,000.00 in 15 years for a special vacation following graduation by making deposits at the end of each six months in an annuity that pays 3.8% compounded semiannually. How much money should you deposit at the end of every six months? How much of the $5,000.00 comes from deposits and how much comes from interest?

Learn how to calculate the semiannual deposits required to reach a $5,000 savings goal in 15 years with 3.8% interest compounded semiannually. This guide covers key concepts in annuities and compound interest, including the formula for future value and periodic deposits. Suitable for high school students in Grades 10-12.

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