Sure, let's break down the problem and solve it step by step.
Given:
- Principal loan amount (P) = R50 000
- Annual interest rate = 26%
- Loan term = 5 years
- The interest is compounded monthly.
First, we need to convert the annual interest rate to a monthly interest rate:
- Annual interest rate (r_annual) = 26%
- Monthly interest rate (r) = [tex]\( \frac{26\%}{12} = \frac{0.26}{12} \approx 0.02167 \)[/tex]
Next, we need to determine the total number of monthly payments (n):
- Duration of the loan (in years) = 5 years
- Number of monthly payments per year = 12 months
- Total number of monthly payments (n) = [tex]\( 5 \times 12 = 60 \)[/tex]
Now we use the formula for calculating the monthly installment (M) of a loan, which is given by the annuity formula:
[tex]\[ M = P \times \frac{r \times (1 + r)^n}{(1 + r)^n - 1} \][/tex]
Plugging in all values we have:
- [tex]\( P = 50000 \)[/tex]
- [tex]\( r \approx 0.02167 \)[/tex]
- [tex]\( n = 60 \)[/tex]
[tex]\[ M = 50000 \times \frac{0.02167 \times (1 + 0.02167)^{60}}{(1 + 0.02167)^{60} - 1} \][/tex]
This calculation will give us the monthly installment.
Performing this operation will give:
[tex]\[ M \approx 1497.02 \][/tex]
So, Simon's monthly installment should be approximately R1497.02.
