You want to purchase a new car in

33

years and expect the car to cost

​$70 comma 00070,000.

Your bank offers a plan with a guaranteed APR of

5.5 %5.5%

if you make regular monthly deposits. How much should you deposit each month to end up with

​$70 comma 00070,000

in

33

​years?

Question content area bottom

Part 1

You should invest

​$enter your response here

each month.

​(Do not round until the final answer. Then round to two decimal places as​ needed.)

To calculate how much you need to deposit each month to accumulate $70,000 in 33 years with an annual interest rate of 5.5%, compounded monthly, we use the **future value of an annuity formula**:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $FV$ is the future value you want, which is $70,000.
- $P$ is the monthly payment (this is what we are solving for).
- $r$ is the monthly interest rate (the annual rate divided by 12).
- $n$ is the total number of payments (number of years times 12).

### Step-by-step process:

1. **Annual interest rate (APR)**: 5.5%, so the monthly interest rate $r$ is:
   \[
   r = \frac{5.5\%}{12} = \frac{0.055}{12} = 0.0045833
   \]

2. **Number of months (n)**: 
   \[
   n = 33 \times 12 = 396 \text{ months}
   \]

3. **Rearrange the formula to solve for $P$**:
   \[
   P = \frac{FV \times r}{(1 + r)^n - 1}
   \]
   
4. Substitute the known values into the formula:
   \[
   P = \frac{70,000 \times 0.0045833}{(1 + 0.0045833)^{396} - 1}
   \]

I'll calculate this now.You should invest approximately **$62.72** each month to accumulate $70,000 in 33 years, with a guaranteed APR of 5.5%.

Do you need further details or have any questions?

Here are 5 related questions:
1. How would the required monthly deposit change if the APR increased to 6%?
2. What would happen if you stopped making monthly deposits after 20 years?
3. How much total interest will you earn over the 33 years?
4. What is the future value if you could contribute $100 per month instead?
5. How does the future value formula change if interest is compounded quarterly?

**Tip:** The higher the interest rate, the smaller the monthly deposit needed to reach your savings goal.

You want to purchase a new car in 33 years and expect the car to cost $70,000. Your bank offers a plan with a guaranteed APR of 5.5% if you make regular monthly deposits. How much should you deposit each month to end up with $70,000 in 33 years?

Learn how to calculate the required monthly deposit to save $70,000 in 33 years with a guaranteed 5.5% APR. Using the future value of an annuity formula, this step-by-step guide helps you solve compound interest problems. Suitable for students in Grades 11-12.

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