The cost of a machine depreciated by rupes 4,752 during the second year and by 4,181.76 during the third year. Calculate :
the cost of the machine at the end of the third year.

Answer

To find the cost of the machine at the end of the third year, we need to follow these steps:

### Step 1: Understand the Depreciation Values
The depreciation values given are:
- Depreciation during the second year: Rs. 4,752
- Depreciation during the third year: Rs. 4,181.76

### Step 2: Calculate the Rate of Depreciation
The depreciation amount decreased from the second year to the third year. We can find the difference in depreciation:
\[ \text{Difference} = 4752 - 4181.76 = 570.24 \]

Next, we can calculate the rate of depreciation based on the second year's depreciation:
\[ \text{Rate of Depreciation} = \left(\frac{570.24}{4752}\right) \times 100 \approx 12\% \]

### Step 3: Assume the Original Cost of the Machine
Let's assume the original cost of the machine is Rs. 100 for easier calculations.

### Step 4: Calculate the Cost After Each Year
1. **After the First Year**:
\[ \text{Cost after 1 year} = 100 - (12\% \text{ of } 100) = 100 - 12 = 88 \]

2. **After the Second Year**:
\[ \text{Cost after 2 years} = 88 - (12\% \text{ of } 88) = 88 - 10.56 = 77.44 \]

3. **After the Third Year**:
\[ \text{Cost after 3 years} = 77.44 - (12\% \text{ of } 77.44) = 77.44 - 9.2928 \approx 68.1472 \]

### Step 5: Calculate the Original Cost of the Machine
Now we can find the original cost of the machine based on the depreciation values:
\[ \text{Original Cost} = 100 \times \frac{4752}{10.56} \approx 45000 \]

### Step 6: Calculate the Cost After Each Year Using the Original Cost
1. **After the First Year**:
\[ \text{Cost after 1 year} = 45000 - (12\% \text{ of } 45000) = 45000 - 5400 = 39600 \]

2. **After the Second Year**:
\[ \text{Cost after 2 years} = 39600 - (12\% \text{ of } 39600) = 39600 - 4752 = 34848 \]

3. **After the Third Year**:
\[ \text{Cost after 3 years} = 34848 - (12\% \text{ of } 34848) = 34848 - 4177.76 = 30666.24 \]

### Final Answer
The cost of the machine at the end of the third year is approximately **Rs. 30,666.24**.

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