A bottle is sold at a loss of 9%. Had it been sold for Rs. 15 more, there would have been a profit of 25/2% (12.5%). Approximately what is the cost price of the bottle?

Introduction / Context:
This question involves two different selling prices for the same bottle, one giving a loss and the other giving a profit. From the difference between these two selling prices and the respective percentages of loss and profit, we can find the cost price. This is a standard type of profit and loss problem, where the same item is hypothetically sold under two different scenarios.

Given Data / Assumptions:
- In the first case, the bottle is sold at a loss of 9%. - In the second case, the bottle would be sold for Rs. 15 more than the first selling price. - In this second case, there would be a profit of 25/2% (that is 12.5%). - The cost price of the bottle is the same in both scenarios. - We need the approximate cost price of the bottle.

Concept / Approach:
Let the cost price be CP. If the bottle is sold at a 9% loss, then the first selling price SP1 = 0.91 * CP. If sold at a profit of 12.5%, then the second selling price SP2 = 1.125 * CP. The problem states that SP2 is Rs. 15 greater than SP1. Therefore, 1.125 * CP = 0.91 * CP + 15. This linear equation in CP can be solved to obtain the cost price. Because the final number is not very neat, an approximate answer is required.

Step-by-Step Solution:
Step 1: Let CP be the cost price of the bottle. Step 2: Selling price at 9% loss: SP1 = 0.91 * CP. Step 3: Selling price at 12.5% profit: SP2 = 1.125 * CP. Step 4: Given that SP2 is Rs. 15 more than SP1. Step 5: So, 1.125 * CP = 0.91 * CP + 15. Step 6: Subtract 0.91 * CP from both sides: (1.125 - 0.91) * CP = 15. Step 7: This gives 0.215 * CP = 15. Step 8: Hence CP = 15 / 0.215. Step 9: CP is approximately equal to 69.77, which we round to about Rs. 70.

Verification / Alternative check:
Take CP as Rs. 70 to check the reasonableness. At a loss of 9%, SP1 would be 70 * 0.91 = Rs. 63.7. At a profit of 12.5%, SP2 would be 70 * 1.125 = Rs. 78.75. The difference SP2 - SP1 is 78.75 - 63.7 = Rs. 15.05, which is extremely close to the given difference of Rs. 15. Since the question asks for an approximate cost price, Rs. 70 is the most suitable choice.

Why Other Options Are Wrong:
Values such as Rs. 55, Rs. 60 or Rs. 65 would produce differences between the two selling prices that are noticeably different from Rs. 15. For example, if CP were Rs. 60, then SP1 would be 54.6 and SP2 would be 67.5, giving a difference of 12.9, which does not match the condition. Only CP close to Rs. 70 generates a difference of about Rs. 15 and satisfies both profit and loss percentages.

Common Pitfalls:
Some students mistakenly treat 25/2% as 25% instead of 12.5%, which leads to a wrong equation. Others forget that the Rs. 15 difference is between two selling prices, not between selling price and cost price. Another common mistake is to round too early in the calculation, which can lead to a noticeably inaccurate cost price. It is better to compute CP accurately first and then round only at the final step when answering an approximate question.

Final Answer:
The approximate cost price of the bottle is about Rs. 70.