We need to find the present value $P$ that will amount to $250,000 if it is invested at 6% compounded semiannually for 10 years.

1. Problem Description

We need to find the present value $P$ that will amount to $250,000 if it is invested at 6% compounded semiannually for 10 years.

2. Solution Steps

The formula for compound interest is:

$A = P(1 + \frac{r}{n})^{nt}$

where:

$A$ is the future value of the investment/loan, including interest

$P$ is the principal investment amount (the present value)

$r$ is the annual interest rate (as a 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 for

We are given:

$A =$250,000$

$r = 0.06$

$n = 2$ (semiannually means twice a year)

$t = 10$ years

We want to find $P$. Rearranging the formula to solve for $P$, we get:

$P = \frac{A}{(1 + \frac{r}{n})^{nt}}$

Plugging in the given values:

$P = \frac{250000}{(1 + \frac{0.06}{2})^{2 \cdot 10}}$

$P = \frac{250000}{(1 + 0.03)^{20}}$

$P = \frac{250000}{(1.03)^{20}}$

$P = \frac{250000}{1.80611123467}$

$P \approx 138418.79$

3. Final Answer

$138418.79$