We need to find the amount of each deposit made at the end of each 6-month period into a sinking fund to discharge a debt of $60,000 in 15 years, given that the interest rate is 6% compounded semiannually.

1. Problem Description

We need to find the amount of each deposit made at the end of each 6-month period into a sinking fund to discharge a debt of $60,000 in 15 years, given that the interest rate is 6% compounded semiannually.

2. Solution Steps

The formula for the periodic payment $PMT$ required to accumulate a future value $FV$ with regular deposits at the end of each period is given by:

$PMT = \frac{FV \times i}{(1 + i)^n - 1}$

Where:

$FV$ is the future value or the target amount to be accumulated.

$i$ is the interest rate per period.

$n$ is the number of periods.

In our case:

FV = \60,000$

The annual interest rate is 6%, compounded semiannually, so the interest rate per period is $i = 6\% / 2 = 0.06 / 2 = 0.03$.

The deposits are made every 6 months for 15 years, so the number of periods is $n = 15 \times 2 = 30$.

Plugging these values into the formula, we get:

PMT = \frac{\60,000 \times 0.03}{(1 + 0.03)^{30} - 1}$

PMT = \frac{\1800}{(1.03)^{30} - 1}$

First, calculate $(1.03)^{30}$:

$(1.03)^{30} \approx 2.42726$

Now, substitute this back into the equation:

PMT = \frac{\1800}{2.42726 - 1}$

PMT = \frac{\1800}{1.42726}$

PMT \approx \1261.106$

Rounding to the nearest cent, we get PMT \approx \1261.11$.

3. Final Answer

$1261.11