Patrick deposited an amount of R18 500 into an account earning r% interest p.a., compounded monthly. After 6 months, his balance was R19 319,48. 6.1.1 Calculate the value of r. 6.1.2 Calculate the effective interest rate.

Given: $P = \text{R}18\,500$, $A = \text{R}19\,319.48$, $n = 6$ months, $m = 12$ (compounded monthly).

$$19319.48 = 18500\left(1 + \frac{r}{12}\right)^{6}$$

$$\frac{19319.48}{18500} = \left(1 + \frac{r}{12}\right)^{6}$$ $$1.044296216 \approx \left(1 + \frac{r}{12}\right)^{6}$$

$$\sqrt[6]{1.044296216} = 1 + \frac{r}{12}$$ $$1.007233333 \approx 1 + \frac{r}{12}$$

$$0.007233333 \approx \frac{r}{12}$$ $$r \approx 0.007233333 \times 12$$ $$r \approx 0.0868$$

To express $r$ as a percentage:

$$r \approx 0.0868 \times 100\%$$ $$\boxed{\text{8.68\%}}$$

Question 6.1.2

Calculate the effective interest rate.

The formula for the effective interest rate is $i_{eff} = \left(1 + \frac{i_{nom}}{m}\right)^m - 1$, where $i_{nom}$ is the nominal annual interest rate and $m$ is the number of compounding periods per year.

$$i_{eff} = (1 + 0.007233333)^{12} - 1$$ $$i_{eff} = (1.007233333)^{12} - 1$$ $$i_{eff} \approx 1.0905 - 1$$ $$i_{eff} \approx 0.0905$$

To express as a percentage:

$$i_{eff} \approx 0.0905 \times 100\%$$ $$\boxed{\text{9.05\%}}$$

Question 6.2.1

The value of the old laptop depreciates annually at a rate of 20% p.a. according to the straight-line method. After how many years will the laptop have a value of R0?

The formula for straight-line depreciation is $A = P(1 - in)$, where $A$ is the future value, $P$ is the present value, $i$ is the annual depreciation rate, and $n$ is the number of years.

Given: $P = \text{R}10\,000$, $i = 20\% = 0.20$, $A = \text{R}0$.

$$0 = 10000(1 - 0.20n)$$

$$0 = 1 - 0.20n$$ $$0.20n = 1$$ $$n = \frac{1}{0.20}$$ $$n = 5$$

The laptop will have a value of R0 after:

$$\boxed{\text{5 years}}$$

Question 6.2.2

Kuda will buy a laptop that costs R20 000. In order to cover the cost price, he made his first monthly deposit into a savings account on 28 February 2019. He will make his 60th monthly deposit on 31 January 2024. The savings account pays interest at 8.7% p.a., compounded monthly. Calculate Kuda's monthly deposit into this account.

This is a future value of an ordinary annuity problem. The formula is $FV = x \frac{[(1+i)^n - 1]}{i}$, where $FV$ is the future value, $x$ is the monthly deposit, $i$ is the monthly interest rate, and $n$ is the number of deposits.

Given: $FV = \text{R}20\,000$, annual interest rate $= 8.7\% = 0.087$, $n = 60$ deposits.

$$i = \frac{0.087}{12} = 0.00725$$

$$(1.00725)^{60} \approx 1.5434805$$ $$\frac{[1.5434805 - 1]}{0.00725} = \frac{0.5434805}{0.00725} \approx 74.962827586$$

$$20000 = x \times 74.962827586$$ $$x = \frac{20000}{74.962827586}$$ $$x \approx 266.79$$

Kuda's monthly deposit is:

$$\boxed{\text{R266.79}}$$

Question 6.3

Tino wins a jackpot of R1 600 000. He invests all of his winnings in a fund that earns interest of 11.2% p.a., compounded monthly. He withdraws R20 000 from the fund at the end of each month. His first withdrawal is exactly 1 month after his initial investment. How many withdrawals of R20 000 will Tino be able to make from this fund?

This is a present value of an ordinary annuity problem. The formula is $PV = x \frac{[1 - (1+i)^{-n}]}{i}$, where $PV$ is the present value, $x$ is the withdrawal amount, $i$ is the monthly interest rate, and $n$ is the number of withdrawals.

Given: $PV = \text{R}1\,600\,000$, $x = \text{R}20\,000$, annual interest rate $= 11.2\% = 0.112$.

$$i = \frac{0.112}{12} = 0.009333333$$

$$\frac{1600000 \times 0.009333333}{20000} = 1 - (1.009333333)^{-n}$$ $$\frac{14933.33328}{20000} = 1 - (1.009333333)^{-n}$$ $$0.746666664 = 1 - (1.009333333)^{-n}$$ $$(1.009333333)^{-n} = 1 - 0.746666664$$ $$(1.009333333)^{-n} = 0.253333336$$

$$-n \log(1.009333333) = \log(0.253333336)$$ $$-n = \frac{\log(0.253333336)}{\log(1.009333333)}$$ $$-n \approx \frac{-0.596321}{-0.004037}$$ $$-n \approx -147.71$$ $$n \approx 147.71$$

Since Tino can only make full withdrawals, we round down to the nearest whole number.
Tino will be able to make:

$$\boxed{\text{147 withdrawals}}$$