Aitana wants $5,000 saved in 5 years to make a down payment on a house. How much money should she invest now at 3.85% compounded semiannually in order to meet her goal?

To determine how much Aitana should invest now to reach her goal of $5,000 in 5 years with an interest rate of 3.85% compounded semiannually, we need to use the formula for compound interest. The formula is:

\[ P = \frac{A}{(1 + \frac{r}{n})^{nt}} \]

Where:
- $ P $ is the present value (the amount to invest now),
- $ A $ is the future value (the amount Aitana wants, which is $5,000),
- $ r $ is the annual interest rate (3.85% or 0.0385),
- $ n $ is the number of times the interest is compounded per year (2 for semiannually),
- $ t $ is the number of years the money is invested (5 years).

Plugging the values into the formula:

\[ P = \frac{5000}{(1 + \frac{0.0385}{2})^{2 \times 5}} \]

Let's calculate the values step-by-step:

1. Calculate the semiannual interest rate: 
   \[ \frac{0.0385}{2} = 0.01925 \]

2. Calculate the number of compounding periods: 
   \[ 2 \times 5 = 10 \]

3. Calculate the compound factor:
   \[ (1 + 0.01925)^{10} = (1.01925)^{10} \]

4. Calculate $ (1.01925)^{10} $:

   Using a calculator:
   \[ (1.01925)^{10} \approx 1.207044 \]

5. Divide the future value by the compound factor:
   \[ P = \frac{5000}{1.207044} \approx 4142.04 \]

So, Aitana should invest approximately $4,142.04 now at 3.85% compounded semiannually to have $5,000 in 5 years.

Do you want more details or have any questions?

Here are 5 more questions you might find useful:
1. How do you calculate compound interest for different compounding frequencies?
2. What is the difference between simple and compound interest?
3. How can you calculate the amount of interest earned over a specific period?
4. How does changing the interest rate affect the final amount in compound interest calculations?
5. What are some practical examples of compound interest in real-life financial planning?

**Tip:** When calculating compound interest, always ensure that the interest rate and the number of compounding periods are consistent (e.g., if the rate is annual, the periods should be per year).

Learn how to calculate compound interest for a $5,000 goal in 5 years at 3.85% interest compounded semiannually. This detailed solution covers the compound interest formula and step-by-step calculations. Suitable for high school students.

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