A manufacturer sold a machine to a wholesale dealer at a profit of $10\%$. The wholesale dealer sold it to a retailer at a profit of $20\%$. While transporting some defect occurred in the machine and hence the retailer sold it at a loss of $5\%$. The customer paid ₹ 627. Find the cost of the machine for the manufacturer. (M.A.T., 2010)

### Concept & Chain Rule in Profit and Loss When an item passes through multiple hands with successive profits and losses, we apply the chain rule of percentages. Final Price = Initial Price $\times$ (1 + Profit$_1\%$) $\times$ (1 + Profit$_2\%$) $\times$ (1 - Loss$_3\%$) $$ SP = CP \times \left( \frac{100 \pm P/L_1}{100} \right) \times \left( \frac{100 \pm P/L_2}{100} \right) \dots $$ ### Step-by-Step Solution * Given: * Profit by manufacturer = $10\%$ * Profit by wholesale dealer = $20\%$ * Loss by retailer = $5\%$ * Final amount paid by customer = ₹ 627 * Let the original cost of the machine for the manufacturer be $x$. * Apply the successive change multipliers: * $x \times \left(1 + \frac{10}{100}\right) \times \left(1 + \frac{20}{100}\right) \times \left(1 - \frac{5}{100}\right) = 627$ * $x \times \frac{110}{100} \times \frac{120}{100} \times \frac{95}{100} = 627$ * $x \times \frac{11}{10} \times \frac{6}{5} \times \frac{19}{20} = 627$ * $x \times \frac{1254}{1000} = 627$ * Solving for $x$: * $x = 627 \times \frac{1000}{1254}$ * $x = \frac{1000}{2} = 500$ ### Exam Strategy & Shortcut Work with percentage multipliers directly as fractions for faster cancellation: $10\% \text{ profit} = \frac{11}{10}$, $20\% \text{ profit} = \frac{6}{5}$, $5\% \text{ loss} = \frac{19}{20}$. Set up the equation $x \times \frac{11 \times 6 \times 19}{10 \times 5 \times 20} = 627$. Notice that $11 \times 19 = 209$, and $209 \times 3 = 627$. So, $x \times \frac{3 \times 2}{1000} = 3 \Rightarrow x = 500$. ### Common Pitfall A common mistake is adding the percentages together ($10 + 20 - 5 = 25\%$) and calculating the profit on the base price. This ignores the compounding nature of successive transactions. ### Final Answer Therefore, the correct answer is **₹ 500**.