what present value P amounts to $270,000 if it is invested at 5%, compounded semiannually, for 17 years? Round to the nearest cent

To find the present value $ P $ that amounts to $270,000 when invested at 5% interest, compounded semiannually for 17 years, we can use the formula for compound interest:

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

Where:
- $ A $ is the amount of money accumulated after n years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the number of years the money is invested or borrowed.

### Given:
- $ A = 270,000 $
- $ r = 0.05 $
- $ n = 2 $ (since it is compounded semiannually)
- $ t = 17 $

### Rearranging the formula to find $ P $:

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

### Plugging in the values:

\[
P = \frac{270,000}{\left(1 + \frac{0.05}{2}\right)^{2 \times 17}}
\]

Calculating step by step:

1. Calculate $ \frac{0.05}{2} = 0.025 $.
2. Calculate $ 1 + 0.025 = 1.025 $.
3. Calculate $ nt = 2 \times 17 = 34 $.
4. Calculate $ 1.025^{34} $.

Now we compute:

\[
1.025^{34} \approx 2.292865
\]

Then:

\[
P = \frac{270,000}{2.292865} \approx 117,708.83
\]

Thus, the present value $ P $ is approximately **$117,708.83**.

### Summary:
The present value $ P $ that amounts to $270,000 at a 5% interest rate compounded semiannually for 17 years is **$117,708.83**.

Do you want more details or have any questions? Here are some related questions you might find interesting:

1. How would the present value change if the interest rate were 6% instead?
2. What would be the future value if the same amount were invested for 20 years at the same rate?
3. How does compounding frequency affect the present value?
4. What is the total interest earned over the 17 years in this investment?
5. How would the present value change if the investment period were reduced to 10 years? 

**Tip:** When calculating present values, always consider the impact of different compounding periods on the final amount.

What present value P amounts to $270,000 if it is invested at 5%, compounded semiannually, for 17 years? Round to the nearest cent.

Discover how to calculate the present value that will amount to $270,000 when invested at a 5% interest rate compounded semiannually for 17 years. Step-by-step solution leading to the present value of approximately $117,708.83.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation