How much would you need to deposit in an account now in order to have $2,000.00 in the account in 9 years if the interest is:

(a) 7.29% simple interest? 


(b) 7.29% compounded monthly?

To determine how much you need to deposit today to reach $2,000 in 9 years, we can calculate this using two different methods: simple interest and compound interest (compounded monthly).

### (a) Simple Interest

The formula for simple interest is:

\[
A = P(1 + rt)
\]

Where:
- $ A $ is the future value ($2,000 in this case),
- $ P $ is the principal (the amount you need to deposit today),
- $ r $ is the annual interest rate (7.29%, or 0.0729),
- $ t $ is the time in years (9 years).

Rearranging the formula to solve for $ P $:

\[
P = \frac{A}{1 + rt}
\]

Now, substitute the known values:

\[
P = \frac{2000}{1 + (0.0729 \times 9)} = \frac{2000}{1 + 0.6561} = \frac{2000}{1.6561} \approx 1,207.61
\]

So, you would need to deposit approximately **$1,207.61** in the account today with simple interest.

---

### (b) Compound Interest (Compounded Monthly)

The formula for compound interest is:

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

Where:
- $ A $ is the future value ($2,000),
- $ P $ is the principal,
- $ r $ is the annual interest rate (0.0729),
- $ n $ is the number of times interest is compounded per year (12 for monthly compounding),
- $ t $ is the time in years (9).

Rearranging the formula to solve for $ P $:

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

Substitute the known values:

\[
P = \frac{2000}{\left(1 + \frac{0.0729}{12}\right)^{12 \times 9}} = \frac{2000}{\left(1 + 0.006075\right)^{108}} = \frac{2000}{(1.006075)^{108}} = \frac{2000}{1.8321}
\]

\[
P \approx 1,091.61
\]

So, you would need to deposit approximately **$1,091.61** in the account today with monthly compounded interest.

---

### Summary:
- (a) With simple interest: Deposit **$1,207.61**.
- (b) With monthly compounded interest: Deposit **$1,091.61**.

Would you like more details or have any questions?

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Here are 5 related questions:
1. What would the future value be with different interest rates?
2. How does the frequency of compounding affect the required deposit?
3. What would happen if the interest rate is compounded annually instead of monthly?
4. How does simple interest compare to compound interest over different time periods?
5. How can inflation affect the value of money over 9 years?

**Tip**: Compound interest grows faster over time because interest is calculated on both the initial deposit and accumulated interest.

How much would you need to deposit in an account now in order to have $2,000.00 in the account in 9 years if the interest is: (a) 7.29% simple interest? (b) 7.29% compounded monthly?

Learn how to calculate the initial deposit needed to reach $2,000 in 9 years with 7.29% interest, using both simple and compound interest formulas. Explore detailed step-by-step solutions, key formulas, and comparisons between simple and compounded monthly interest.

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