A sinking fund is established to discharge a debt of $60,000 in 15 years. Deposits are made at the end of each 6-month period, and interest is paid at a rate of 6%, compounded semiannually. We need to find the amount of each deposit.

1. Problem Description

A sinking fund is established to discharge a debt of $60,000 in 15 years. Deposits are made at the end of each 6-month period, and interest is paid at a rate of 6%, compounded semiannually. We need to find the amount of each deposit.

2. Solution Steps

The formula for the payment $PMT$ of a sinking fund is:

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

Where:

$FV$ is the future value (the amount to be accumulated).

$i$ is the interest rate per period.

$n$ is the number of periods.

In this problem, we have:

$FV =$60,000$

The interest rate is 6% per year, compounded semiannually. So, the interest rate per period is $i = \frac{6\%}{2} = 0.03$.

The time period is 15 years, and deposits are made semiannually. So, the number of periods is $n = 15 \times 2 = 30$.

Plugging these values into the formula, we get:

$PMT = \frac{60000 \cdot 0.03}{(1 + 0.03)^{30} - 1}$

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

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

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

$PMT \approx 1261.101$

Rounding to the nearest cent, we get $1261.

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3. Final Answer

$1261.10