Let's solve the problem step-by-step:
### Step 1: Determine the Total Amount of Interest Payable on the Loan
Nosipho's uncle agrees to lend her R7,000 at a 5% per annum simple interest rate. The time period for the loan is 2 years.
The formula to calculate simple interest (I) is:
[tex]\[ I = \frac{P \cdot r \cdot t}{100} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (R7,000),
- [tex]\( r \)[/tex] is the annual interest rate (5%),
- [tex]\( t \)[/tex] is the time in years (2 years).
Plugging in the numbers:
[tex]\[ I = \frac{7000 \cdot 5 \cdot 2}{100} \][/tex]
[tex]\[ I = \frac{70000}{100} \][/tex]
[tex]\[ I = 700 \][/tex]
Hence, the total amount of interest payable on the loan is R700.
### Step 2: Calculate How Much Will Be Paid Back in Total After 2 Years
To find the total amount (A) to be paid back after 2 years, we add the interest to the principal amount.
The formula for the total amount after covering interest is:
[tex]\[ A = P + I \][/tex]
We already know:
- [tex]\( P = 7000 \)[/tex]
- [tex]\( I = 700 \)[/tex]
Therefore:
[tex]\[ A = 7000 + 700 \][/tex]
[tex]\[ A = 7700 \][/tex]
Thus, the total amount to be paid back after 2 years is R7,700.
### Step 3: Justify the Monthly Installment Amount
Nosipho's uncle proposed that she pay back the loan in equal monthly installments over the 2 years. The total amount to be paid back is R7,700 over a period of 2 years, which corresponds to 24 months.
To find the monthly installment (E), we divide the total amount by the number of months:
[tex]\[ E = \frac{A}{t \cdot 12} \][/tex]
where:
- [tex]\( A \)[/tex] is the total amount to be paid back (R7,700),
- [tex]\( t \)[/tex] is the time in years (2 years),
- [tex]\( 12 \)[/tex] is the number of months in a year.
Plugging in the numbers:
[tex]\[ E = \frac{7700}{2 \cdot 12} \][/tex]
[tex]\[ E = \frac{7700}{24} \][/tex]
[tex]\[ E \approx 320.83 \][/tex]
Thus, the monthly installment amount is R320.83.
By breaking down the total amount (R7,700) over the 24 months during the 2 years, Nosipho would indeed need to pay R320.83 each month. The calculated result confirms that her uncle's proposed monthly installment amount is correct.
