A computer system cost $3,000 new. It depreciates at 20% p.a. After the 2nd year, a further $500 worth of equipment is added. What is the value of the total system after 5 years?




To find the value of the total system after 5 years, we can break down the problem into two main parts: the depreciation of the original system and the addition of new equipment.

### Step 1: Depreciation of the Original System
The system depreciates at 20% per annum. This means each year, the value of the system is reduced by 20% of its value from the previous year.

The value of the system after $ n $ years can be calculated using the formula:

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

where:
- $ V_0 = 3000 $ is the initial value of the system,
- $ r = 0.20 $ is the depreciation rate,
- $ n $ is the number of years.

#### After 2 years:
\[
V_2 = 3000 \times (1 - 0.20)^2 = 3000 \times 0.8^2 = 3000 \times 0.64 = 1920
\]

### Step 2: Adding New Equipment
After the 2nd year, $500 worth of new equipment is added. So the total value of the system at the beginning of the 3rd year is:

\[
\text{Total value after 2 years and adding new equipment} = 1920 + 500 = 2420
\]

### Step 3: Depreciation Over the Next 3 Years
Now, we calculate the depreciation of this new total value over the next 3 years.

#### After 3 years:
\[
V_3 = 2420 \times (1 - 0.20) = 2420 \times 0.8 = 1936
\]

#### After 4 years:
\[
V_4 = 1936 \times (1 - 0.20) = 1936 \times 0.8 = 1548.8
\]

#### After 5 years:
\[
V_5 = 1548.8 \times (1 - 0.20) = 1548.8 \times 0.8 = 1239.04
\]

### Final Answer
The value of the total system after 5 years is **$1239.04**.

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. What would be the value after 5 years if no additional equipment was added after the 2nd year?
2. How would the value change if the depreciation rate was 25% instead of 20%?
3. What is the total depreciation amount over the 5 years?
4. How would the calculation change if the equipment was added at the beginning of the 2nd year?
5. What would be the final value if $1,000 worth of equipment was added instead of $500?
6. How does compound interest differ from depreciation in this context?
7. What is the difference in value if the system had a depreciation rate of 10% instead of 20%?
8. How much would the system be worth after 10 years under the same conditions?

**Tip:** When calculating depreciation, always check if any new value is added at any point, as it can significantly affect the outcome.

Explore how to calculate the total value of a computer system after 5 years considering depreciation at 20% per annum and the addition of $500 worth of equipment after the 2nd year. Understand the depreciation formula and its application, suitable for advanced high school students.

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