a) The focal date as the present day means we need to calculate the present day value of the two debts. The overdue one has accumulate 6 months’ interest: 3000(1+0.09/2)=$3135. We need to work out the present day value (PV) of the future debt: 5000/(1+0.09×3/2)=$4405.29.
The total of the two debts at the present day is $7762.44.
In 4 months’ time the $3000 debt rises to 3000(1+0.09×5/6)=$3225, while the adjusted $5000 debt rises to 4405.29(1+0.09/3)=$4537.44, making a total of $7762.44. Then a payment of $x is made and we have to find x.
After another 6 months the $3000 debt rises to 3000(1+0.09×16/12)=$3360 while the $5000 debt rises to 4405.29(1+0.09×5/6)=$4735.68, making a total of $8095.68. Also the payment of $x will rise to x(1+0.09/2)=$1.045x. Then the second payment of $x is made to settle both debts. Assuming that this does not gain interest, the total deductions from the two debts comes to x+1.045x=2.045x.
Therefore, 8095.68-2.045x=0 and x=8095.68/2.045=$3958.77. So the equal payments come to $3958.77.
b) The focal date is the due date of the $5000 debt.
When the first payment of $x is made, it is 14 months early so the value of prepayment is based on the value 14 months earlier=5000/(1+0.09×14/12)=$4524.89. When the second payment is made this has risen to $4728.51. The commitment on the $3000 debt is the same as in (a). So on the date of the second payment the total debt is $8088.51. At settlement we have 8088.51-2.045x=0, so x=8088.51/2.045=$3955.26.