Question

Question 6

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 af 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?


Tom owes Jerry: R 1 300.00


Jerry owes Tom: R 3 3830.30


Tom owes Jerry: R 4 032.86


Jerry owes Tom: R 4 032.86


None of the above

Answer 1

To determine how much Tom and Jerry will owe each other now (present value), we need to calculate the present value of the amounts owed by both parties.

### Step 1: Calculate the present value of what Tom owes Jerry.

**Amount owed by Tom (Future Value)**: R 19,000
**Interest rate**: 12% per annum, compounded monthly
**Number of years**: 8 years

The formula for present value (PV) is:

\[
PV = \frac{FV}{(1 + r/n)^{nt}}
\]

Where:
- \( FV \) = future value
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of compounding periods per year
- \( t \) = number of years

**Values**:
- \( FV = 19,000 \)
- \( r = 0.12 \)
- \( n = 12 \)
- \( t = 8 \)

Calculating \( PV \):

\[
PV = \frac{19000}{(1 + 0.12/12)^{12 \times 8}} = \frac{19000}{(1 + 0.01)^{96}} = \frac{19000}{(1.01)^{96}}
\]

Calculating \( (1.01)^{96} \):

\[
(1.01)^{96} \approx 2.21964
\]

Now calculating the present value:

\[
PV \approx \frac{19000}{2.21964} \approx 8,554.77
\]

### Step 2: Calculate the present value of what Jerry owes Tom.

**Amount owed by Jerry (Future Value)**: R 17,700
**Interest rate**: 9% per annum, compounded quarterly
**Number of years**: 5 years

Using the same present value formula:

**Values**:
- \( FV = 17,700 \)
- \( r = 0.09 \)
- \( n = 4 \)
- \( t = 5 \)

Calculating \( PV \):

\[
PV = \frac{17700}{(1 + 0.09/4)^{4 \times 5}} = \frac{17700}{(1 + 0.0225)^{20}} = \frac{17700}{(1.0225)^{20}}
\]

Calculating \( (1.0225)^{20} \):

\[
(1.0225)^{20} \approx 1.51459
\]

Now calculating the present value:

\[
PV \approx \frac{17700}{1.51459} \approx 11,691.12
\]

### Step 3: Compare the present values.

- Present value of what Tom owes Jerry: R 8,554.77
- Present value of what Jerry owes Tom: R 11,691.12

### Conclusion

Jerry owes Tom R 11,691.12 now, while Tom owes Jerry R 8,554.77 now.

To cancel out the debts, Jerry would need to pay the difference between the amounts owed:

\[
Payment = 11,691.12 - 8,554.77 = 3,136.35
\]

This means Jerry should make a payment of approximately R 3,136.35 to Tom.

Since this amount is not explicitly shown in the options provided, the answer is:

**None of the above.**