Calculate what the replacement value of the new machine will be. What will the quarterly payment be that will make provision for the new machine if the scrap value of the old machine is put down as a deposit on the new machine?

The current cost of the machine ($PV$) is R278 000.
The appreciation rate ($i$) is 7.5% per annum, or 0.075.
The time period ($n$) is 10 years.

Substitute the values:

The replacement value of the new machine will be R573 066.75.

3.2.2
To determine the quarterly payment for the sinking fund, we first calculate the amount needed after considering the scrap value, then use the future value of an annuity formula.

The replacement value (from 3.2.1) is R573 066.75.
The scrap value of the old machine is R25 000.
Amount needed in sinking fund ($FV_{sinking}$) = Replacement value - Scrap value

The future value needed ($FV_{sinking}$) is R548 066.75.
The interest rate ($i$) is 3.8% per quarter, or 0.038.
The number of quarters ($n$) is $10 \text{ years} \times 4 \text{ quarters/year} = 40$ quarters.

Substitute the values:

The quarterly payment will be R6 031.47.

3.3
To find the equivalent annual compound interest rate, we first calculate the future value using simple interest, then use that value to solve for the compound interest rate.

Assume a principal ($P$) of R1 for simplicity.
The simple interest rate ($i_{simple}$) is 22% per annum, or 0.22.
The time period ($n$) is 3 years.

Let $i_{compound}$ be the annual compound interest rate.

Convert to a percentage:

The annual rate at compound interest will be 18.44%.