Sonia wants to have $13,000 in 10 years. Use Table 11-2 to calculate how much she should invest now (in $) at 6% interest, compounded semiannually in order to reach this goal. (Round your answer to the nearest cent.)
$

To determine how much Sonia should invest now to have $13,000 in 10 years, we use the formula for present value with compound interest. The formula is:

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

Where:
- $ P $ is the present value (the amount Sonia needs to invest now),
- $ A $ is the future value (the amount Sonia wants in the future, $13,000),
- $ r $ is the annual interest rate (6% or 0.06),
- $ n $ is the number of compounding periods per year (since it's semiannual, $ n = 2 $),
- $ t $ is the time in years (10 years).

Let’s calculate the amount Sonia should invest.

\[
P = \frac{13000}{(1 + \frac{0.06}{2})^{2 \times 10}}
\]

\[
P = \frac{13000}{(1 + 0.03)^{20}} = \frac{13000}{(1.03)^{20}}
\]

Now, we’ll compute $ (1.03)^{20} $ and find $ P $.

Sonia should invest approximately **$7,197.78** now in order to have $13,000 in 10 years at 6% interest, compounded semiannually.

Would you like more details on this calculation or have any questions?

Here are 5 related questions to consider:
1. What if the interest rate were 8% instead of 6%?
2. How would the calculation change if the interest were compounded quarterly?
3. What if Sonia wanted to reach $13,000 in only 5 years instead of 10?
4. How much total interest will Sonia earn by the end of the 10 years?
5. What would the formula look like for continuous compounding?

**Tip:** Semiannual compounding means interest is added twice a year, resulting in slightly more earnings compared to annual compounding.

Sonia wants to have $13,000 in 10 years. Use Table 11-2 to calculate how much she should invest now (in $) at 6% interest, compounded semiannually in order to reach this goal. (Round your answer to the nearest cent.)

Learn how to calculate the present value needed to reach $13,000 in 10 years at a 6% interest rate compounded semiannually. This problem uses the compound interest formula and is suitable for students in Grades 10-12.

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