How much money should be deposited today in an account that earns 4 % compounded semiannually so that it will accumulate to $ 11,000 in three​ years?

To determine how much money should be deposited today in an account that earns 4% interest compounded semiannually, we use the formula for the present value of a compound interest account:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

Where:
- $ PV $ is the present value (the amount to be deposited today).
- $ FV $ is the future value (the amount you want to accumulate, which is $11,000).
- $ r $ is the annual interest rate (4%, or 0.04).
- $ n $ is the number of times interest is compounded per year (semiannually, so $ n = 2 $).
- $ t $ is the number of years the money is invested (3 years).

Let's plug in the values:

\[
PV = \frac{11,000}{\left(1 + \frac{0.04}{2}\right)^{2 \times 3}}
\]

First, calculate the term inside the parentheses:

\[
1 + \frac{0.04}{2} = 1 + 0.02 = 1.02
\]

Then raise it to the power of $ 2 \times 3 = 6 $:

\[
1.02^6 \approx 1.12616
\]

Finally, divide the future value by this amount to find the present value:

\[
PV = \frac{11,000}{1.12616} \approx 9,765.30
\]

So, approximately **$9,765.30** should be deposited today.

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

### Related Questions:
1. What is the future value of $9,765.30 deposited for three years at 4% compounded semiannually?
2. How does the compounding frequency affect the amount to be deposited?
3. What would be the present value if the interest rate was compounded annually instead of semiannually?
4. How much would need to be deposited if the interest rate was 5% compounded semiannually?
5. How much would be needed if the desired future value was $15,000 instead of $11,000?
6. What happens to the required deposit if the time period is reduced to 2 years?
7. How would continuous compounding affect the amount to be deposited?
8. How does changing the interest rate to 3% affect the required deposit amount?

**Tip:** The more frequently interest is compounded, the less you need to deposit today to reach your desired future value.

Learn how much money should be deposited today in an account earning 4% interest compounded semiannually to accumulate $11,000 in three years. Get a step-by-step solution using the compound interest formula. Suitable for grades 10-12.

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