Question
Mrs. Jacobs borrowed R 5 000 from a cash loans institution. It was agreed by the respective parties that the term of this loan will be 3 years and that the debt will be repaid in a single payment at the end of this term. Initially interest on this loan was calculated at an effective rate of 18% per annum. After 24 months, however, the cash loans institution changed their policy concerning the interest rate so that for the remainder of the term interest has to calculate at a rate of 15% per annum compounded monthly. What amount of interest will Mrs. Thatcher pay on this loan?
R 7 049.06
R 3 081.49
R 1 962
None of the above
R 7 049.06
R 3 081.49
R 1 962
None of the above
Answers
GPT-4o mini
To calculate the total amount of interest Mrs. Jacobs will pay on the loan, we need to handle both the initial interest rate for the first 24 months and the new interest rate for the remaining 12 months.
### Step 1: Calculate the amount due after the first 24 months at an effective rate of 18% per annum.
The formula for the future value (\(FV\)) with simple interest is:
\[
FV = P(1 + rt)
\]
Where:
- \(P\) = Principal amount (R 5,000)
- \(r\) = Interest rate (18% per annum = 0.18)
- \(t\) = Time in years (2 years)
Substituting the values:
\[
FV = 5000(1 + 0.18 \times 2) = 5000(1 + 0.36) = 5000 \times 1.36 = R 6,800
\]
### Step 2: Now, calculate the amount due after the next 12 months at a rate of 15% per annum compounded monthly.
The formula for the future value with compound interest is:
\[
FV = P(1 + \frac{r}{n})^{nt}
\]
Where:
- \(r\) = Interest rate (15% per annum = 0.15)
- \(n\) = Number of times interest is compounded per year (12)
- \(t\) = Time in years (1 year)
After 24 months, since the amount is now R 6,800, we have:
\[
FV = 6800 \left(1 + \frac{0.15}{12}\right)^{12 \times 1}
\]
Calculating:
1. First, calculate \(\frac{0.15}{12} = 0.0125\)
2. Compute \((1 + 0.0125) = 1.0125\)
3. Raise this to the power of 12:
\[
1.0125^{12} \approx 1.16075
\]
Now, substituting this back:
\[
FV \approx 6800 \times 1.16075 \approx R 7,897.10
\]
### Step 3: Calculate the total interest paid.
The amount Mrs. Jacobs will owe at the end of 3 years is approximately R 7,897.10.
The total interest paid would be:
\[
\text{Total Interest} = \text{Amount Owed} - \text{Principal} = 7897.10 - 5000 = R 2,897.10
\]
Since this amount is not listed among the choices given (R 7,049.06, R 3,081.49, R 1,962), the answer is:
**None of the above.**
### Step 1: Calculate the amount due after the first 24 months at an effective rate of 18% per annum.
The formula for the future value (\(FV\)) with simple interest is:
\[
FV = P(1 + rt)
\]
Where:
- \(P\) = Principal amount (R 5,000)
- \(r\) = Interest rate (18% per annum = 0.18)
- \(t\) = Time in years (2 years)
Substituting the values:
\[
FV = 5000(1 + 0.18 \times 2) = 5000(1 + 0.36) = 5000 \times 1.36 = R 6,800
\]
### Step 2: Now, calculate the amount due after the next 12 months at a rate of 15% per annum compounded monthly.
The formula for the future value with compound interest is:
\[
FV = P(1 + \frac{r}{n})^{nt}
\]
Where:
- \(r\) = Interest rate (15% per annum = 0.15)
- \(n\) = Number of times interest is compounded per year (12)
- \(t\) = Time in years (1 year)
After 24 months, since the amount is now R 6,800, we have:
\[
FV = 6800 \left(1 + \frac{0.15}{12}\right)^{12 \times 1}
\]
Calculating:
1. First, calculate \(\frac{0.15}{12} = 0.0125\)
2. Compute \((1 + 0.0125) = 1.0125\)
3. Raise this to the power of 12:
\[
1.0125^{12} \approx 1.16075
\]
Now, substituting this back:
\[
FV \approx 6800 \times 1.16075 \approx R 7,897.10
\]
### Step 3: Calculate the total interest paid.
The amount Mrs. Jacobs will owe at the end of 3 years is approximately R 7,897.10.
The total interest paid would be:
\[
\text{Total Interest} = \text{Amount Owed} - \text{Principal} = 7897.10 - 5000 = R 2,897.10
\]
Since this amount is not listed among the choices given (R 7,049.06, R 3,081.49, R 1,962), the answer is:
**None of the above.**