Finance. A person wishes to have $ 16 comma 700 cash for a new car 3 years from now. How much should be placed in an account​ now, if the account pays 5.3 % annual interest​ rate, compounded​ weekly?
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Part 1
​$
  
enter your response here ​(Round to the nearest​ dollar.)

To find out how much should be placed in an account now (present value) to reach $16,700 in 3 years with a 5.3% annual interest rate compounded weekly, we use the compound interest formula:

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

Where:
- $ A $ = future value ($16,700)
- $ P $ = present value (the amount to be invested now)
- $ r $ = annual interest rate (5.3% or 0.053)
- $ n $ = number of times interest is compounded per year (weekly compounding means 52 times per year)
- $ t $ = time in years (3 years)

We want to solve for $ P $. Rearranging the formula:

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

Let's plug in the values and calculate.The amount that should be placed in the account now is **$14,246** (rounded to the nearest dollar).

Would you like further details or have any other questions?

Here are 5 related questions for exploration:
1. What if the interest were compounded daily instead of weekly?
2. How would the present value change if the interest rate were 6% instead of 5.3%?
3. What is the effect of compounding frequency on the final amount?
4. How much would need to be invested for a future value of $20,000 instead of $16,700?
5. What is the formula for continuous compounding, and how would it apply here?

**Tip**: The more frequently interest is compounded, the higher the future value will be, all else being equal.

A person wishes to have $16,700 cash for a new car 3 years from now. How much should be placed in an account now, if the account pays 5.3% annual interest rate, compounded weekly?

Discover how to calculate the amount you need to invest now to reach $16,700 in 3 years with a 5.3% interest rate compounded weekly. This problem explores key mathematical concepts, including the compound interest formula, suitable for students in Grades 10-12.

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