To determine how much money should be deposited today, we use the compound interest formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

where:

• $A = 9000$ (the future value)
• $P$ = initial deposit (what we are solving for)
• $r = 7\% = 0.07$ (annual interest rate)
• $n = 2$ (compounded semiannually)
• $t = 3$ years

Step 1: Plug in the values

$9000 = P \left(1 + \frac{0.07}{2}\right)^{2(3)}$

$9000 = P \left(1.035\right)^6$

$9000 = P \times 1.23144$

Step 2: Solve for $P$

$P = \frac{9000}{1.23144}$

$P \approx 7308.95$

Final Answer:

The amount that should be deposited today is
$\mathbf{\$7308.95}$

Would you like a breakdown of another part of the process or a different scenario?

Related Questions:

1. What if the interest were compounded quarterly instead of semiannually?
2. How does changing the interest rate to 8% affect the initial deposit?
3. How much would the deposit be if the account was compounded annually?
4. How much would the deposit be if the time period was extended to 5 years?
5. What would be the accumulated amount if $7308.95 was left for 5 years instead of 3?

Tip:

Always round up when dealing with financial calculations to ensure enough funds are deposited!