A debt of $3,000 due six months ago and another $5,000 due in 18 months are to be settled by two equal payments, one at the end of four months and the other at the end of ten months. Find the size of the payments using

a) the present as the focal date,
b) the end of 18 months as the focal date,
assuming money is worth 9% per annum simple interest.

To find the size of the payments using the present as the focal date, we need to calculate the present value of the two debts.

a) The present value formula for a debt is:

Present Value = Future Value / (1 + interest rate * time)

First, let's calculate the present value of the $3,000 debt due six months ago. The time is 6 months, and the interest rate is 9% per annum (or 4.5% for six months).

Present Value = $3,000 / (1 + 0.045 * 6) = $3,000 / 1.27 = $2,362.20

Next, let's calculate the present value of the $5,000 debt due in 18 months. The time is 18 months, and the interest rate is 9% per annum (or 13.5% for 18 months).

Present Value = $5,000 / (1 + 0.135 * 18) = $5,000 / 3.43 = $1,458.71

Now, let's add the present values of the two debts:

Total Present Value = $2,362.20 + $1,458.71 = $3,820.91

To settle the total present value of $3,820.91 with two equal payments, we divide it by 2:

Size of Payments = $3,820.91 / 2 = $1,910.46 (rounded to two decimal places)

So, using the present as the focal date, the size of the payments is approximately $1,910.46.

b) To find the size of the payments using the end of 18 months as the focal date, we need to calculate the future value of the two debts.

b) The future value formula for a debt is:

Future Value = Present Value * (1 + interest rate * time)

First, let's calculate the future value of the $3,000 debt due six months ago. The time is 12 months (6 months remaining until the end of 18 months), and the interest rate is 9% per annum (or 4.5% for six months).

Future Value = $2,362.20 * (1 + 0.045 * 12) = $2,362.20 * 1.54 = $3,631.07

Next, let's calculate the future value of the $5,000 debt due in 18 months. The time is 0 months (no time remaining until the end of 18 months), and the interest rate is 9% per annum.

Future Value = $1,458.71 * (1 + 0.09 * 0) = $1,458.71 * 1 = $1,458.71

Now, let's add the future values of the two debts:

Total Future Value = $3,631.07 + $1,458.71 = $5,089.78

To settle the total future value of $5,089.78 with two equal payments, we divide it by 2:

Size of Payments = $5,089.78 / 2 = $2,544.89 (rounded to two decimal places)

So, using the end of 18 months as the focal date, the size of the payments is approximately $2,544.89.