The value of a machine depreciates at the rate of 25% each year. If the difference between its values at the end of the third and the fourth year is Rs. 17,199. then what is the value of the machine at the end of the first year?

Options

A. Rs. 1,32,208
B. Rs. 1,63,072
C. Rs. 1,83,456
D. Rs. 1,22,304

Answer

To solve the problem step by step, we will follow the depreciation formula and the information given in the question.

### Step 1: Understand the Depreciation Rate
The machine depreciates at a rate of 25% each year. This means that at the end of each year, the value of the machine is 75% of its value at the beginning of that year. In fractional terms, this is represented as:
\[ \text{Value after 1 year} = \text{Initial Value} \times \left(1 - \frac{25}{100}\right) = \text{Initial Value} \times \frac{3}{4} \]

### Step 2: Calculate the Value at the End of Each Year
Let the initial value of the machine be \( P \).
- At the end of the first year:
\[ \text{Value after 1 year} = P \times \frac{3}{4} \]

- At the end of the second year:
\[ \text{Value after 2 years} = P \times \left(\frac{3}{4}\right)^2 \]

- At the end of the third year:
\[ \text{Value after 3 years} = P \times \left(\frac{3}{4}\right)^3 \]

- At the end of the fourth year:
\[ \text{Value after 4 years} = P \times \left(\frac{3}{4}\right)^4 \]

### Step 3: Set Up the Equation Based on Given Information
According to the problem, the difference between the values at the end of the third and fourth years is Rs. 17,199:
\[ \text{Value after 4 years} - \text{Value after 3 years} = 17199 \]
Substituting the values we calculated:
\[ P \times \left(\frac{3}{4}\right)^4 - P \times \left(\frac{3}{4}\right)^3 = 17199 \]

### Step 4: Factor Out Common Terms
Factoring out \( P \times \left(\frac{3}{4}\right)^3 \):
\[ P \times \left(\frac{3}{4}\right)^3 \left( \frac{3}{4} - 1 \right) = 17199 \]
This simplifies to:
\[ P \times \left(\frac{3}{4}\right)^3 \left(-\frac{1}{4}\right) = 17199 \]
Rearranging gives:
\[ P \times \left(\frac{3}{4}\right)^3 \times \left(-\frac{1}{4}\right) = 17199 \]

### Step 5: Solve for \( P \)
Now, we can isolate \( P \):
\[ P \times \left(\frac{3}{4}\right)^3 = 17199 \times -4 \]
Calculating \( -4 \times 17199 \):
\[ P \times \left(\frac{3}{4}\right)^3 = -68796 \]
Now, we can calculate \( \left(\frac{3}{4}\right)^3 = \frac{27}{64} \):
\[ P \times \frac{27}{64} = 68796 \]
Thus,
\[ P = 68796 \times \frac{64}{27} \]
Calculating this gives:
\[ P = 163072 \]

### Step 6: Calculate the Value at the End of the First Year
Now, we need to find the value of the machine at the end of the first year:
\[ \text{Value after 1 year} = P \times \frac{3}{4} = 163072 \times \frac{3}{4} = 122304 \]

### Final Answer
The value of the machine at the end of the first year is Rs. 122304.

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