How much money should be deposited today in an account that earns 7% compounded semiannually so that it will accumulate to $9000 in three​ years?

We are solving for the **present value** (the amount to be deposited today) in a compound interest setting. Here’s how we can calculate it:

### Formula for Compound Interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:  
- $A$ = Future value (accumulated amount) = 9000  
- $P$ = Present value (amount to be deposited) = ?  
- $r$ = Annual interest rate = 7% or 0.07  
- $n$ = Number of times compounded per year = 2 (semiannually)  
- $t$ = Time in years = 3  

### Rearranging the formula to solve for $P$:
\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]

### Substituting the values:
\[
P = \frac{9000}{\left(1 + \frac{0.07}{2}\right)^{2 \cdot 3}}
\]
\[
P = \frac{9000}{\left(1 + 0.035\right)^6}
\]
\[
P = \frac{9000}{1.035^6}
\]

### Calculating the result:
\[
1.035^6 \approx 1.2309
\]
\[
P = \frac{9000}{1.2309} \approx 7311.74
\]

### Final Answer:
The amount to be deposited today is approximately **$7311.74**.

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Would you like further clarification on any part of the solution?  

### 5 Related Questions:
1. What happens if the compounding frequency changes to quarterly?
2. How much interest is earned over the 3 years in this scenario?
3. How does changing the interest rate to 8% affect the deposit amount?
4. What is the formula for continuous compounding?
5. How can the total accumulated value change if the time increases to 5 years?

### Tip:
Always double-check whether the interest rate is compounded annually, semiannually, or quarterly, as it significantly affects the final value.

Learn how to calculate the present value of an investment to accumulate $9000 over three years with a 7% interest rate compounded semiannually. The solution provides a detailed step-by-step explanation of using the compound interest formula, suitable for high school students in Grades 10-12.

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