8.1 A farmer bought a tractor for R980 000 - NSC Mathematics - Question 10 - 2021 - Paper 1

Using the formula for compound interest:

$A = P(1 + \frac{r}{n})^{nt}$

where:

• $A$ = future value = R116 253.50,
• $P$ = present value = R75 000,
• $r$ = annual interest rate = 0.068,
• $n$ = number of times interest is compounded per year (quarterly = 4),
• $t$ = number of years.

Rearranging the formula:

$t = \frac{n}{r} \log\left(\frac{A}{P}\right)$

Substituting the values will yield the time required.

For monthly deposits with compound interest, we apply the future value of an annuity formula:

$A = PMT \frac{(1 + r)^n - 1}{r}$

Where:

• $A$ = future value = R450,000,
• $PMT$ = monthly deposit,
• $r$ = monthly interest rate = \frac{0.0835}{12},
• $n$ = total deposits over the period.

Calculate $n$ based on the time between 31 July 2013 and 30 June 2018. Substitute the known values to find $PMT$.

The outstanding balance on a loan can be calculated with the loan amortization formula:

$B = P(1 + r)^n - PMT \frac{(1 + r)^n - 1}{r}$

Where:

• $B$ = balance,
• $P$ = principal (loan amount),
• $r$ = monthly interest rate = \frac{0.12}{12},
• $n$ = total number of payments made until 2039,
• $PMT$ = monthly installment = R11 085.85.

Calculate $n$ based on 21 years and substitute to find the balance.

Total interest paid can be calculated using:

$\text{Total Interest} = (PMT \cdot n) - P$

Where:

• $PMT$ = monthly payment,
• $n$ = total number of payments over 21 years.

Substitute the known values into this equation to find the total interest paid over the specified period.