A department has ordered 

99 

new computers at a cost of 

​$22842284 

each. The order will not be delivered for 

99 

months. What amount could the department deposit in a special 

99​-month 

CD paying 

3.683.68​% 

compounded monthly to have enough to pay for the machines at time of​ delivery?

Question content area bottom

Part 1

The department should deposit 

​$enter your response here 

into the CD.

​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

To solve this problem, we need to determine how much the department should deposit today in a 99-month Certificate of Deposit (CD) paying 3.68% interest, compounded monthly, in order to have enough to pay for the computers when the 99 months are over.

### Step 1: Compute the Total Cost of the Computers
Each computer costs \$22,842,284, and the department is ordering 99 computers, so:

\[
\text{Total Cost} = 99 \times 22,842,284
\]

### Step 2: Use the Future Value Formula for Compound Interest
We need to determine how much to deposit now, so we’ll use the formula for the future value of a compounded interest investment:

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

Where:
- $A$ is the future value (the amount needed at the end of the 99 months),
- $P$ is the present value (the amount to be deposited),
- $r$ is the annual interest rate (as a decimal),
- $n$ is the number of times the interest is compounded per year (since it's monthly, $n = 12$),
- $t$ is the number of years (99 months is $ \frac{99}{12} = 8.25$ years).

### Step 3: Rearrange the Formula to Solve for $P$
We are looking for the present value $P$, so we rearrange the formula:

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

Now let's compute this using the provided numbers and determine the exact amount.The department should deposit \$1,670,031,554.89 into the 99-month CD to have enough to pay for the computers at the time of delivery.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How does the compound interest formula differ for different compounding periods (e.g., quarterly vs monthly)?
2. What is the impact of increasing the interest rate on the initial deposit needed?
3. How would the initial deposit change if the number of months were reduced to 60?
4. How is the present value calculation used in other financial decisions like loans or mortgages?
5. How would the solution change if the interest rate was compounded daily instead of monthly?

**Tip:** When working with compound interest problems, make sure to convert the interest rate into a decimal and adjust for the compounding frequency (monthly, quarterly, etc.).

A department has ordered 99 new computers at a cost of $22,842,284 each. The order will not be delivered for 99 months. What amount could the department deposit in a special 99-month CD paying 3.68% compounded monthly to have enough to pay for the machines at the time of delivery?

Learn how to calculate the amount to deposit in a 99-month CD at 3.68% interest to pay for 99 computers costing $22,842,284 each. This problem covers key finance concepts, including compound interest and future value formulas. Suitable for students in Grades 11-12 and college-level finance courses.

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