A machine depreciates in value each year at the rate of 10% of its previous value. However, every second year there is some maintenance work so that in that particular year, depreciation is only 5% of its previous value. If at the end of the fourth year, the value of the machine stands at Rs. 1,46,205, then find the value of the machine at the start at the first year?

Options

A. Rs. 1,90,000
B. Rs. 2,00,000
C. Rs. 1,95,000
D. Rs. 2,10,000

Answer

To find the initial value of the machine at the start of the first year, we can work backwards from the final value after four years. The machine depreciates at different rates each year, so we need to account for that.

Let's denote the initial value of the machine as \( x \).

### Step-by-Step Solution:

1. **Understanding Depreciation Rates**:
- For the first year, the machine depreciates at 10%.
- For the second year, due to maintenance, it depreciates at 5%.
- For the third year, it again depreciates at 10%.
- For the fourth year, it depreciates at 5%.

2. **Calculating Value After Each Year**:
- After Year 1: The value of the machine will be \( x \times (1 - 0.10) = x \times 0.90 \).
- After Year 2: The value will be \( (x \times 0.90) \times (1 - 0.05) = x \times 0.90 \times 0.95 \).
- After Year 3: The value will be \( (x \times 0.90 \times 0.95) \times (1 - 0.10) = x \times 0.90 \times 0.95 \times 0.90 \).
- After Year 4: The value will be \( (x \times 0.90 \times 0.95 \times 0.90) \times (1 - 0.05) = x \times 0.90 \times 0.95 \times 0.90 \times 0.95 \).

3. **Final Value Equation**:
- We know that at the end of the fourth year, the value of the machine is Rs. 1,46,205.
- Therefore, we can set up the equation:
\[ x \times 0.90 \times 0.95 \times 0.90 \times 0.95 = 1,46,205 \]

4. **Simplifying the Equation**:
- Calculate the product of the depreciation factors:
\[ 0.90 \times 0.95 \times 0.90 \times 0.95 = 0.90^2 \times 0.95^2 = 0.81 \times 0.9025 = 0.73125 \]
- So, we have:
\[ x \times 0.73125 = 1,46,205 \]

5. **Finding the Initial Value**:
- Now, solve for \( x \):
\[ x = \frac{1,46,205}{0.73125} \]
- Calculate \( x \):
\[ x \approx 2,00,000 \]

### Final Answer:
The initial value of the machine at the start of the first year is approximately Rs. 2,00,000.

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