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

To determine the time required for R75,000 to grow to R116,253.50 with interest compounded quarterly, we can use the formula:

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

Where:

• $A$ = final amount (R116,253.50)
• $P$ = principal amount (R75,000)
• $r$ = annual interest rate (6.8% or 0.068)
• $n$ = number of compounding periods per year (4 for quarterly)
• $t$ = number of years.

Rearranging the formula to solve for $t$:

$t = \frac{\log(A/P)}{n \cdot \log(1 + \frac{r}{n})}$

Substituting the known values:

$t = \frac{\log(116253.50/75000)}{4 \cdot \log(1 + \frac{0.068}{4})}$

Calculating:

1. Calculate $A/P$: $$\frac{116253.50}{75000} \approx 1.550$

2. Calculate $1 + \frac{0.068}{4} \approx 1.017$

3. Using logarithms in a calculator: $t \approx \frac{\log(1.550)}{4 \cdot \log(1.017)} \approx \frac{0.191}{4 \cdot 0.0074} \approx 6.388.$

Thus, it will take approximately 6.39 years for the amount to accrue.

To find the monthly deposit amount, we can use the future value of an annuity formula:

$FV = PMT \times \frac{(1 + r/n)^{nt} - 1}{r/n}$

Where:

• $FV$ = future value (R450,000)
• $PMT$ = monthly deposit amount (unknown)
• $r$ = annual interest rate (8.35% or 0.0835)
• $n$ = number of compounding periods per year (12 for monthly)
• $t$ = number of years between 31 July 2013 and 30 June 2018, which is about 5 years.

Rearranging for $PMT$:

$PMT = \frac{FV \cdot (r/n)}{(1 + r/n)^{nt} - 1}$

Substituting the values:

$PMT = \frac{450000 \cdot (0.0835/12)}{(1 + 0.0835/12)^{60} - 1}$

Calculating using a calculator: $PMT \approx \frac{450000 \cdot 0.00695833}{1.550694 - 1} \approx \frac{3133.24}{0.550694} \approx 5683.40$

Thus, the monthly deposit amount is approximately R5,683.40.

To calculate the balance of the loan after 21 years, we use the formula:

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

Where:

• $B$ = balance outstanding
• $P$ = loan amount (R1,500,000 - R450,000)
• $r$ = annual interest rate (12% or 0.12)
• $n$ = number of payments per year (12)
• $PMT$ = monthly payment (R11,058.85)
• $t$ = total time in years (25).

First, calculate the principal amount borrowed: $P = 1500000 - 450000 = 1050000$

Now substituting into the formula: $B = 1050000(1 + 0.12/12)^{12 imes 21} - 11058.85 \times \frac{(1 + 0.12/12)^{12 imes 21} - 1}{0.12/12}$

Calculating, we find:

• $B \approx 1050000 \cdot 9.6504 - 11058.85 \cdot 145.505\approx 1012394 - 1608930.79 \approx 425145$.

Thus, the outstanding balance is approximately R425,145.

To find the total interest paid over 21 years, we can calculate:

1. Total amount paid in instalments:

\text{ where } n = 12, \text{ hence:}\ \text{Total Payments} = 11058.85 \cdot 12 \cdot 21$$ 2. Total amount paid: $$\text{Total Payments} = 11058.85 \cdot 252\approx 278,345.42$$ 3. Total interest paid: $$\text{Total Interest} = ext{Total Payments} - ext{Initial Loan Amount}$$. Assuming the initial loan amount remains as calculated before (R1,050,000), the total interest: $$\text{Total Interest} = 278,345.42 - 1,050,000 \approx 278,345.42$$. Thus, approximately R278,345.42 will be paid in interest.