Question

How much money should be deposited today in an account that earns 6.68% compounded semiannually so that it will accumulate to $23,900 in 27 years?

Use this formula:
PV=FV/(1+r/n)^nt




Enter the dollar amount rounded up to the nearest cent.

Your Answer:

Answer 1

To find the present value (PV) that needs to be deposited, we can use the formula provided:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

where:
- \( FV \) is the future value, which is $23,900.
- \( r \) is the annual interest rate (as a decimal), which is 6.68% or 0.0668.
- \( n \) is the number of times the interest is compounded per year, which is 2 (since it's compounded semiannually).
- \( t \) is the number of years the money is invested, which is 27.

Now, we need to substitute these values into the formula:

1. Calculate \( \frac{r}{n} \):
\[
\frac{r}{n} = \frac{0.0668}{2} = 0.0334
\]

2. Calculate \( nt \):
\[
nt = 2 \times 27 = 54
\]

3. Now calculate \( (1 + \frac{r}{n})^{nt} \):
\[
(1 + 0.0334)^{54}
\]
First, calculate \( 1 + 0.0334 = 1.0334 \).

Then calculate \( 1.0334^{54} \):
\[
1.0334^{54} \approx 5.0594 \quad \text{(using a calculator)}
\]

4. Now substitute these values back into the PV formula:
\[
PV = \frac{23900}{5.0594}
\]
Calculate this:
\[
PV \approx 4714.21
\]

Therefore, the amount that should be deposited today is **$4714.21.**