Question
Tom owes Jerry a sum of money that will amount to R 19 000 at an interest rate of 12% per annum, compounded monthly, after 8 years. Jerry in turn owes Tom a sum of money that will amount to R 17 700 at a rate of 9% per annum, compounded quarterly, after 5 years. They agree to cancel their respective liabilities by making at present a single payment. Who should make this payment and what should this payment amount to?
Answers
GPT-4o mini
To determine who should make the present payment to settle the debts between Tom and Jerry, we need to calculate the present values of both debts.
### Step 1: Calculate Present Value of Tom's Debt to Jerry
Tom owes Jerry R 19,000 after 8 years at an interest rate of 12% per annum compounded monthly. The formula for the present value (PV) is:
\[
PV = \frac{FV}{(1 + r/n)^{nt}}
\]
Where:
- \( FV = 19000 \) (future value)
- \( r = 0.12 \) (annual interest rate)
- \( n = 12 \) (number of times interest is compounded per year)
- \( t = 8 \) (time in years)
Plugging in the values:
\[
PV = \frac{19000}{(1 + 0.12/12)^{12 \times 8}} = \frac{19000}{(1 + 0.01)^{96}} = \frac{19000}{(1.01)^{96}}
\]
Calculating \( (1.01)^{96} \):
\[
(1.01)^{96} \approx 2.21964
\]
Now calculating the present value:
\[
PV = \frac{19000}{2.21964} \approx 8,550.87
\]
### Step 2: Calculate Present Value of Jerry's Debt to Tom
Jerry owes Tom R 17,700 after 5 years at an interest rate of 9% per annum compounded quarterly. Using the same formula:
\[
PV = \frac{FV}{(1 + r/n)^{nt}}
\]
Where:
- \( FV = 17700 \)
- \( r = 0.09 \) (annual interest rate)
- \( n = 4 \) (number of times interest is compounded per year)
- \( t = 5 \) (time in years)
Plugging in the values:
\[
PV = \frac{17700}{(1 + 0.09/4)^{4 \times 5}} = \frac{17700}{(1 + 0.0225)^{20}} = \frac{17700}{(1.0225)^{20}}
\]
Calculating \( (1.0225)^{20} \):
\[
(1.0225)^{20} \approx 1.50662
\]
Now calculating the present value:
\[
PV = \frac{17700}{1.50662} \approx 11,754.16
\]
### Step 3: Compare Present Values and Determine Payment
- Tom's debt to Jerry: R 8,550.87
- Jerry's debt to Tom: R 11,754.16
To cancel the debts, Jerry owes more. The difference between the debts is:
\[
11,754.16 - 8,550.87 \approx 3,203.29
\]
### Conclusion
Jerry should make a payment of approximately R 3,203.29 to Tom to settle their debts.
### Step 1: Calculate Present Value of Tom's Debt to Jerry
Tom owes Jerry R 19,000 after 8 years at an interest rate of 12% per annum compounded monthly. The formula for the present value (PV) is:
\[
PV = \frac{FV}{(1 + r/n)^{nt}}
\]
Where:
- \( FV = 19000 \) (future value)
- \( r = 0.12 \) (annual interest rate)
- \( n = 12 \) (number of times interest is compounded per year)
- \( t = 8 \) (time in years)
Plugging in the values:
\[
PV = \frac{19000}{(1 + 0.12/12)^{12 \times 8}} = \frac{19000}{(1 + 0.01)^{96}} = \frac{19000}{(1.01)^{96}}
\]
Calculating \( (1.01)^{96} \):
\[
(1.01)^{96} \approx 2.21964
\]
Now calculating the present value:
\[
PV = \frac{19000}{2.21964} \approx 8,550.87
\]
### Step 2: Calculate Present Value of Jerry's Debt to Tom
Jerry owes Tom R 17,700 after 5 years at an interest rate of 9% per annum compounded quarterly. Using the same formula:
\[
PV = \frac{FV}{(1 + r/n)^{nt}}
\]
Where:
- \( FV = 17700 \)
- \( r = 0.09 \) (annual interest rate)
- \( n = 4 \) (number of times interest is compounded per year)
- \( t = 5 \) (time in years)
Plugging in the values:
\[
PV = \frac{17700}{(1 + 0.09/4)^{4 \times 5}} = \frac{17700}{(1 + 0.0225)^{20}} = \frac{17700}{(1.0225)^{20}}
\]
Calculating \( (1.0225)^{20} \):
\[
(1.0225)^{20} \approx 1.50662
\]
Now calculating the present value:
\[
PV = \frac{17700}{1.50662} \approx 11,754.16
\]
### Step 3: Compare Present Values and Determine Payment
- Tom's debt to Jerry: R 8,550.87
- Jerry's debt to Tom: R 11,754.16
To cancel the debts, Jerry owes more. The difference between the debts is:
\[
11,754.16 - 8,550.87 \approx 3,203.29
\]
### Conclusion
Jerry should make a payment of approximately R 3,203.29 to Tom to settle their debts.