The value of a machine depreciates at the rate of 10% every year on a fixed percentage basis. The machine was purchased 3 years ago. If its present value is Rs. 8,748, what was its purchase price, in rupees, when it was bought?

Introduction / Context:
This problem deals with depreciation, where an asset such as a machine decreases in value by a fixed percentage every year. Depreciation is very common in real-life accounting and financial mathematics, and here it is applied at a constant rate of 10% per year. We are given the current value of the machine after 3 years and must determine its original purchase price. This involves working with repeated percentage reductions and understanding the concept of compound depreciation.

Given Data / Assumptions:
- Annual depreciation rate = 10% of the value at the beginning of each year.- Time elapsed since purchase = 3 years.- Present value of the machine after 3 years = Rs. 8,748.- Depreciation is applied in a multiplicative way each year (compound depreciation).- We need to find the machine's original purchase price.

Concept / Approach:
When an asset depreciates at a fixed percentage rate per year, its value after n years is obtained by multiplying the original value by (1 - rate)^n. Here, the depreciation rate is 10%, so the value multiplies each year by (1 - 0.10) = 0.90. After 3 years, the factor becomes 0.90^3. We write the present value as original value multiplied by 0.90^3 and then solve for the original value by division. This is exactly similar to compound interest calculations, but in reverse and using a negative rate.

Step-by-Step Solution:
Step 1: Let the purchase price of the machine be P rupees.Step 2: Depreciation rate is 10% per year, so the value each year is multiplied by (1 - 0.10) = 0.90.Step 3: After 3 years, the value becomes P * (0.90)^3.Step 4: The present value after 3 years is given as Rs. 8,748.Step 5: Therefore, P * (0.90)^3 = 8,748.Step 6: Compute 0.90^3 = 0.9 * 0.9 * 0.9 = 0.81 * 0.9 = 0.729.Step 7: So we have P * 0.729 = 8,748.Step 8: Solve for P: P = 8,748 / 0.729.Step 9: 8,748 / 0.729 = 12,000.Thus, the purchase price P is Rs. 12,000.

Verification / Alternative check:
We can verify by applying depreciation forward from Rs. 12,000.Year 1: Value = 12,000 * 0.90 = 10,800.Year 2: Value = 10,800 * 0.90 = 9,720.Year 3: Value = 9,720 * 0.90 = 8,748.This matches the given present value exactly, so the purchase price of Rs. 12,000 is confirmed as correct.

Why Other Options Are Wrong:
- 10000: After 3 years at 10% depreciation, 10,000 * 0.9^3 = 7,290, which is less than 8,748.- 14000: 14,000 * 0.9^3 = 10,206, which is greater than 8,748.- 16000: 16,000 * 0.9^3 = 11,664, much higher than the given present value.

Common Pitfalls:
- Treating depreciation as simple subtraction each year instead of multiplying by 0.90 successively.- Using 1 - 3 * 0.10 instead of (1 - 0.10)^3, which ignores the compounding effect.- Rounding too early in the division 8,748 / 0.729, leading to approximate values.- Confusing appreciation (compound interest) with depreciation and using 1 + 0.10 instead of 1 - 0.10.

Final Answer:
The purchase price of the machine was Rs. 12,000.