Raghunandan sells a machine for Rs. 45 lakh at a loss. If instead he had sold the machine for Rs. 55 lakh, his gain would have been nine times the amount of his original loss. What is the cost price of the machine in lakhs of rupees?

Introduction / Context:
This problem involves two hypothetical selling prices of the same machine: one resulting in a loss and the other in a gain. What makes the question interesting is that the gain in the second scenario is specified as nine times the loss in the first scenario. Such questions are common in aptitude exams because they require you to form and solve a linear equation based on profit and loss relationships. The cost price remains the central unknown that connects both cases and is determined using the condition given about the relative sizes of loss and gain.

Given Data / Assumptions:
- In the first scenario, machine is sold for SP1 = Rs. 45 lakh at a loss. - In the second scenario, it would be sold for SP2 = Rs. 55 lakh at a gain. - The gain in the second scenario would be 9 times the loss in the first scenario. - The cost price (CP) of the machine is the same in both cases and is to be found.

Concept / Approach:
Let CP denote the cost price in lakhs. In the first case, loss = CP - 45 (because CP is greater than 45). In the second case, gain = 55 - CP (because CP is less than 55 there). The condition says that gain in the second case is nine times the loss in the first case: 55 - CP = 9 * (CP - 45). This is a simple linear equation in CP, which we can expand and solve. Once CP is known, we can verify that it indeed produces the stated relationship between loss and gain.

Step-by-Step Solution:
Let CP = x lakhs. First scenario: Sold at Rs. 45 lakh, so loss = x - 45 (since x > 45). Second scenario: Sold at Rs. 55 lakh, so gain = 55 - x. Given: gain in second case = 9 * loss in first case. Therefore, 55 - x = 9 * (x - 45). Expand the right-hand side: 55 - x = 9x - 405. Bring all terms to one side: 55 + 405 = 9x + x. So, 460 = 10x. Hence, x = 460 / 10 = 46. Therefore, CP = Rs. 46 lakh.

Verification / Alternative check:
Loss at SP1 = CP - SP1 = 46 - 45 = Rs. 1 lakh. Gain at SP2 = SP2 - CP = 55 - 46 = Rs. 9 lakh. The condition says gain is nine times the loss: 9 lakh = 9 * 1 lakh, which is true. Therefore, CP = 46 lakh perfectly satisfies the given condition.

Why Other Options Are Wrong:
- If CP were 54 lakh, the loss at 45 lakh would be 9 lakh and gain at 55 lakh would be 1 lakh, reversing the required ratio. - CP = 60 lakh or CP = 38 lakh would produce either only loss or only gain in both cases, which contradicts the question statement. - CP = 50 lakh fails the equation 55 - CP = 9(CP - 45) and does not give the gain as nine times the loss.

Common Pitfalls:
Mistaking which part is loss and which is gain can lead to reversing the equation and an incorrect answer. Another frequent error is setting 55 - CP = 9 * (45 - CP) instead of 9 * (CP - 45), which changes the sign of the loss. Some students try to average 45 and 55 directly, which ignores the crucial condition involving the ratio between gain and loss.

Final Answer:
The cost price of the machine is Rs. 46 lakhs.