Question

Franklin owes the following amounts to the same person: $16,000 due today, $11,500 due in 1¼ years, $17,000 due in 2¾ years, and $15,000 due in 4¼ years. He wants to make a single payment of $56,500 instead. Using an interest rate of 8% compounded quarterly, in how many months from today should this payment be made? Round answer to zero decimal places.

Answer 1

Franklin should make a single payment of $56,500 in approximately 19 months from today to pay off his outstanding debts.

Step-by-step explanation:

To find out how many months from today Franklin should make a single payment of $56,500, we need to calculate the future value of his outstanding debts at an interest rate of 8% compounded quarterly. First, let's find the future value of each individual debt using the formula A = P(1 + r/n)^(nt), where A is the future value, P is the principal amount, r is the interest rate, n is the number of compounding periods per year, and t is the number of years.

1. For $16,000 due today:
   A = 16000(1 + 0.08/4)^(4 * 0) = $16,000
2. For $11,500 due in 1¼ years:
   A = 11500(1 + 0.08/4)^(4 * 1.25) = $12,459.22
3. For $17,000 due in 2¾ years:
   A = 17000(1 + 0.08/4)^(4 * 2.75) = $19,845.11
4. For $15,000 due in 4¼ years:
   A = 15000(1 + 0.08/4)^(4 * 4.25) = $19,494.11

Now let's calculate the future value of the total debts:

Total future value = $16,000 + $12,459.22 + $19,845.11 + $19,494.11 = $67,798.44

To determine the number of months it will take to reach the payment of $56,500, we can use the formula A = P(1 + r/n)^(nt), solving for t. Rearranging the formula, we get:

t = (1/n) * (log(A/P) / log(1 + r/n))

Plugging in the values P = $67,798.44, A = $56,500, r = 0.08, and n = 4 (quarterly compounding), we get:

t = (1/4) * (log(56500/67798.44) / log(1 + 0.08/4)) = 19.53 months

Therefore, Franklin should make the single payment of $56,500 in approximately 19 months from today.