A special depreciation ruling was obtained from the Taxation Office on a particular piece of scientific apparatus. For the first four years, it depreciates at 32% p.a., and for the second four years, it depreciates at 22% p.a. Find the total percentage decrease in value.

To find the total percentage decrease in the value of the scientific apparatus over the eight years, we need to calculate the value at the end of both the first and second periods of depreciation, using the respective rates, and then determine the overall decrease.

Let's break the problem down step by step.

### Step 1: Depreciation Formula

The value of an asset after a certain period of depreciation can be modeled by the formula:

\[
V_n = V_0 \times (1 - r)^n
\]

Where:
- \( V_n \) is the value after \( n \) years.
- \( V_0 \) is the initial value.
- \( r \) is the depreciation rate (in decimal form).
- \( n \) is the number of years.

### Step 2: First Four Years at 32% Depreciation Rate

For the first four years, the depreciation rate is 32% per annum. The depreciation factor is:

\[
1 - 0.32 = 0.68
\]

After four years, the value will be:

\[
V_4 = V_0 \times 0.68^4
\]

### Step 3: Next Four Years at 22% Depreciation Rate

For the next four years, the depreciation rate is 22% per annum. The depreciation factor is:

\[
1 - 0.22 = 0.78
\]

The value at the end of the next four years (total eight years) will be:

\[
V_8 = V_4 \times 0.78^4
\]

Substituting \( V_4 = V_0 \times 0.68^4 \) into the equation:

\[
V_8 = V_0 \times 0.68^4 \times 0.78^4
\]

### Step 4: Total Depreciation

Now we calculate the total percentage decrease in value over the eight years. The percentage of the final value relative to the initial value is:

\[
\frac{V_8}{V_0} = 0.68^4 \times 0.78^4
\]

The total percentage decrease is:

\[
\text{Total decrease} = \left( 1 - \frac{V_8}{V_0} \right) \times 100
\]

Now let’s compute this value.The total percentage decrease in value of the scientific apparatus over the eight-year period is approximately **92.09%**.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the total depreciation change if the initial depreciation rate was higher?
2. What if depreciation was compounded quarterly instead of annually? How would that affect the calculation?
3. How does reducing the depreciation rate after a certain period benefit tax calculations?
4. Can the final value of the apparatus be predicted accurately with inflation included?
5. What are the tax implications of depreciation in terms of net profits for businesses?

**Tip:** Depreciation reduces the book value of an asset but does not involve actual cash flow, impacting accounting and tax reporting differently from cash expenses.

Learn how to calculate the total percentage decrease in value of an asset that depreciates at 32% for the first four years and 22% for the next four years. This detailed solution includes key depreciation concepts and formulas, suitable for students in Grades 10-12.

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