The value of a machine depreciates at the rate of 12 percent per annum. It was purchased three years ago. Its present value is ₹ 29644.032. What was the purchase price of the machine?

### Concept & Formula This is a successive depreciation problem where we are given the final depreciated value and need to find the initial purchase price. It acts exactly like reverse compound interest. The formula for compound depreciation is: $$ P_{final} = P_{initial} \times \left(1 - \frac{R}{100}\right)^n $$ ### Step-by-Step Solution * **Given:** Present Value ($P_{final}$) = $29644.032$, Rate of Depreciation ($R$) = $12\%$, Time ($n$) = $3$ years. * **Fractional Multiplier:** A depreciation of $12\%$ means $88\%$ of the value remains each year. As a decimal, the multiplier is $0.88$. As a fraction, it is $\frac{88}{100} = \frac{22}{25}$. * **Set up the equation:** $$ 29644.032 = P_{initial} \times (0.88)^3 $$ * **Isolate the initial price:** $$ P_{initial} = \frac{29644.032}{(0.88)^3} $$ * **Simplify the calculation:** Instead of calculating exact heavy decimals, convert to fractions: $$ P_{initial} = 29644.032 \times \left(\frac{25}{22}\right)^3 $$ $$ P_{initial} = 29644.032 \times \frac{15625}{10648} $$ * Divide $29644.032$ by $10648$: $29644.032 \div 10648 \approx 2.784$ * Multiply by $15625$: $$ 2.784 \times 15625 = 43500 $$ ### Exam Strategy & Shortcut **Approximation Method:** Calculating $0.88^3$ exactly takes too long. Use approximation. $0.88$ is close to $0.9$. $0.9^3 = 0.729$. If the multiplier was $0.729$, the initial price would be roughly $29644 \div 0.729 \approx 40600$. Since the actual multiplier ($0.88^3 \approx 0.681$) is smaller than $0.729$, the original price must be slightly larger than $40600$. Looking at the options, ₹ 43500 is the only logical fit in that range. ### Common Pitfall Avoid trying to calculate the exact decimal value of $(0.88)^3 \times P = 29644.032$ by brute force arithmetic, as the chances of making a multiplication error with the trailing `.032` decimal are extremely high. Rely on fractions or strategic approximation. ### Final Answer **Therefore, the correct answer is ₹ 43500.**