1. A mainframe computer whose cost is $$ \mathrm{K} 220,000 $$ will depreciate to a scrap value of K 12000 in 5 years. (a) If the reducing balance method of depreciation is used, find the depreciation rate. (b) What is the book value of the computer at the end of the third year? (c) how much more would the book value be at the end of the third year if the straight line method of depreciation had been used? 2. A company purchases a machine for $$ k 20,000 $$, which has a useful life of 10 years. If the machine depreciates at a rate of $$ 10 \% $$ per year, what is the depreciation expense for the first year? 3. A building is purchased for k500,000 and has a useful life of 20 years. If the building depreciates at a rate of $$ 5 \% $$ per year, what is the depreciation expense for the first year?

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1. **Mainframe Computer**

The computer costs $$220000$$ and its scrap value after 5 years is $$12000$$.

**(a) Depreciation Rate using the Reducing Balance Method**

In the reducing balance method, the book value after $$n$$ years is given by
   $$
   \text{Book Value} = \text{Cost} \times (1 - r)^n
   $$
   where $$r$$ is the depreciation rate. For 5 years:
   $$
   220000 \times (1 - r)^5 = 12000
   $$
   Solve for $$(1 - r)^5$$:
   $$
   (1 - r)^5 = \frac{12000}{220000} = \frac{12}{220} = \frac{3}{55}
   $$
   Taking the fifth root of both sides:
   $$
   1 - r = \sqrt[5]{\frac{3}{55}}
   $$
   Approximating numerically,
   $$
   \frac{3}{55} \approx 0.054545, \quad \sqrt[5]{0.054545} \approx 0.56
   $$
   Thus,
   $$
   r \approx 1 - 0.56 = 0.44 \quad \text{or} \quad 44\%
   $$

**(b) Book Value at the End of the Third Year**

The book value after 3 years is:
   $$
   \text{Book Value at 3 years} = 220000 \times (1-0.44)^3 = 220000 \times 0.56^3
   $$
   Calculate $$0.56^3$$:
   $$
   0.56^2 = 0.3136, \quad 0.56^3 \approx 0.175616
   $$
   Therefore,
   $$
   \text{Book Value} \approx 220000 \times 0.175616 \approx 38635.52
   $$
   So, the book value is approximately $$38636$$.

**(c) Difference if the Straight Line Method Were Used**

For the straight line method, the annual depreciation is computed by:
   $$
   \text{Annual Depreciation} = \frac{\text{Cost} - \text{Scrap Value}}{\text{Life}} = \frac{220000 - 12000}{5} = \frac{208000}{5} = 41600
   $$
   The book value after 3 years would then be:
   $$
   \text{Book Value (SL)} = 220000 - 3 \times 41600 = 220000 - 124800 = 95200
   $$
   The difference between the book values using straight line and reducing balance methods at the end of the third year is:
   $$
   95200 - 38636 \approx 56564
   $$
2. **Machine Depreciation**

The machine is purchased for $$20000$$ and depreciates at a rate of $$10\%$$ per year. For the first year, using the reducing balance approach (or simply a first-year rate):
   $$
   \text{Depreciation Expense} = 20000 \times 0.10 = 2000
   $$

3. **Building Depreciation**

The building is purchased for $$500000$$ and depreciates at a rate of $$5\%$$ per year. The depreciation expense for the first year is:
   $$
   \text{Depreciation Expense} = 500000 \times 0.05 = 25000
   $$
1. **Mainframe Computer** - **(a) Depreciation Rate:** 44% - **(b) Book Value at End of 3 Years:** Approximately $$38,636$$ - **(c) Difference with Straight Line Method:** The book value would be $$56,564$$ higher if the straight line method were used. 2. **Machine Depreciation** - **First Year Depreciation Expense:** $$2,000$$ 3. **Building Depreciation** - **First Year Depreciation Expense:** $$25,000$$

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