Raj sells a machine for Rs 51 lakh at a loss. If instead he had sold the same machine for Rs 60 lakh, his gain would have been eight times his earlier loss. What is the cost price of the machine (in lakh rupees)?

Introduction / Context:
This question checks understanding of profit and loss when the same article is sold at two different prices, once at a loss and once at a profit. The relationship between these two situations is given in terms of how many times one loss or gain is compared to the other. Such questions are common in aptitude tests to see if the candidate can translate a verbal condition into an algebraic equation involving cost price and selling price.

Given Data / Assumptions:

• First selling price (SP1) is Rs 51 lakh and this sale is at a loss.
• Second selling price (SP2) is Rs 60 lakh and this sale would be at a profit.
• The gain at Rs 60 lakh is eight times the loss at Rs 51 lakh.
• We denote the cost price by CP in lakh rupees.

Concept / Approach:
Loss is calculated as CP minus SP when CP is greater than SP. Profit is calculated as SP minus CP when SP is greater than CP. The key statement is that the profit at the higher selling price equals eight times the loss at the lower selling price. This gives a simple linear equation in the cost price. Solving this equation yields the unknown cost price of the machine.

Step-by-Step Solution:
Step 1: Let CP be the cost price in lakhs.Step 2: At SP1 = 51, loss = CP - 51.Step 3: At SP2 = 60, gain = 60 - CP.Step 4: According to the question, gain at Rs 60 lakh is eight times the loss at Rs 51 lakh.Step 5: Therefore, write the equation: 60 - CP = 8 * (CP - 51).Step 6: Expand the right side: 60 - CP = 8CP - 408.Step 7: Bring CP terms to one side: 60 + 408 = 8CP + CP, so 468 = 9CP.Step 8: Thus CP = 468 / 9 = 52.Step 9: So the cost price of the machine is Rs 52 lakh.

Verification / Alternative check:
We can quickly verify this cost price. If CP is 52, then loss at SP1 (51) is 52 - 51 = 1 lakh. At SP2 (60), profit is 60 - 52 = 8 lakhs. Indeed, the profit of 8 lakhs is exactly eight times the loss of 1 lakh. Thus the condition in the question is satisfied, confirming that the cost price of Rs 52 lakh is correct.

Why Other Options Are Wrong:
Option A (Rs 59 lakhs) gives loss 8 and gain 1, which does not match the condition that gain be eight times the loss. Option C (Rs 66.375 lakhs) makes SP1 and SP2 both losses, contradicting the statement that there is a gain at Rs 60 lakh. Option D (Rs 45 lakhs) makes both transactions profitable instead of one loss and one gain. Only option B produces the correct ratio of profit to loss as described.

Common Pitfalls:
Candidates often confuse which expression represents profit and which represents loss, leading to reversed signs in the equation. Another mistake is to apply the eight times relation on absolute selling prices instead of on the monetary gain or loss amount. Also, some students directly set up the equation without checking whether CP is between 51 and 60, which is necessary to have loss in one case and profit in the other. Carefully defining variables before forming the equation avoids these mistakes.

Final Answer:
The cost price of the machine is Rs 52 lakhs.