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 %

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

70,000 

in 

3 ​years?

Question content area bottom Part 1 You should invest ​$enter your response hereeach month. (Do not round until the final answer. Then round to two decimal places as​ needed.)

To solve this problem, we can use the future value of a series formula to calculate how much you need to deposit monthly. The formula for the future value of a series is:

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

Where:
- $FV$ is the future value (in this case, $70,000),
- $P$ is the monthly deposit,
- $r$ is the monthly interest rate (APR divided by 12),
- $n$ is the total number of deposits (months).

### Step 1: Convert the APR to a monthly rate
The annual percentage rate (APR) is 5.5%, or 0.055. To get the monthly interest rate, divide it by 12:

\[
r = \frac{0.055}{12} = 0.0045833
\]

### Step 2: Calculate the number of months
Since the investment period is 3 years, the number of months is:

\[
n = 3 \times 12 = 36 \, \text{months}
\]

### Step 3: Use the future value formula
We know that the future value ($FV$) is $70,000, and now we can rearrange the formula to solve for $P$ (the monthly deposit):

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

Substitute the values into the formula:

\[
P = \frac{70,000}{\frac{(1 + 0.0045833)^{36} - 1}{0.0045833}}
\]

Now, I'll calculate the result.You should invest approximately **$1792.88** each month to accumulate $70,000 in 3 years, with an APR of 5.5%.

Would you like more details on this or have any further questions?

Here are 5 related questions to explore:
1. How does the interest rate affect the monthly deposit amount?
2. What if the APR was 6% instead of 5.5%?
3. How would the required monthly deposit change if you wanted to save $100,000 instead of $70,000?
4. What is the total interest earned over the 3-year period?
5. How would the result change if you made quarterly deposits instead of monthly?

**Tip:** Increasing the deposit frequency (e.g., from annual to monthly) results in more compounding opportunities and could help you reach your goal faster.

You want to purchase a new car in 3 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 3 years?

Find out how much you need to deposit monthly to save $70,000 in 3 years with an APR of 5.5%. This step-by-step guide uses the future value of a series formula and explores interest rates and compounding, perfect for high school students in Grades 10-12.

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