Rajeev sells a machine for Rs 55 lakhs at a loss. If instead he had sold it for Rs 67 lakhs, his gain would have been five times the former loss. What is the cost price of the machine in lakhs of rupees?

Introduction / Context:
This problem tests your understanding of how profit and loss relate to cost price and selling price in two hypothetical scenarios. Rajeev sells the same machine at two different prices: in one case he incurs a loss, and in another case he earns a gain that is a multiple of that loss. The goal is to set up an equation using the relationship between these two situations to find the unknown cost price.

Given Data / Assumptions:

• First selling price (SP1) is Rs 55 lakhs, which results in a loss.
• Second selling price (SP2) is Rs 67 lakhs, which results in a gain.
• The gain at SP2 is five times the loss at SP1.
• The cost price (CP) of the machine is the same for both scenarios.
• We need CP in lakhs of rupees.

Concept / Approach:
Loss is CP minus SP when SP is less than CP. Gain is SP minus CP when SP is greater than CP. Let the loss at Rs 55 lakhs be L. Then L = CP - 55. The gain at Rs 67 lakhs is 67 - CP. Given that gain equals five times the loss, we write 67 - CP = 5 * (CP - 55). This equation in CP can be solved to obtain the cost price.

Step-by-Step Solution:
Let CP = x lakhs. Loss at SP1 (55 lakhs) is L = x - 55. Gain at SP2 (67 lakhs) is G = 67 - x. Given G = 5L, so 67 - x = 5(x - 55). Expand the right side: 67 - x = 5x - 275. Bring like terms together: 67 + 275 = 5x + x, so 342 = 6x. Thus x = 342 / 6 = 57 lakhs.

Verification / Alternative check:
If CP is 57 lakhs, loss at SP1 is 57 - 55 = 2 lakhs. Gain at SP2 is 67 - 57 = 10 lakhs. The gain of 10 lakhs is exactly five times the loss of 2 lakhs, which matches the condition given. So CP = 57 lakhs is fully consistent.

Why Other Options Are Wrong:
A cost price of 65, 78, or 49 lakhs would not satisfy the relationship that gain at 67 lakhs is five times the loss at 55 lakhs. Checking each quickly shows that the resulting gains and losses do not match the required multiple.

Common Pitfalls:
Many test takers incorrectly write the equation with reversed signs or treat gain and loss as percentages when the question uses absolute rupee values. Another common error is to try to average the two selling prices, which does not use the five times relationship and produces a wrong answer.

Final Answer:
The cost price of the machine is Rs 57 lakhs.