Savita sold an item for Rs. 6,500 and incurred a loss of 20% on its cost price. At what price should she have sold the same item in order to earn a profit of 20% instead?

Introduction / Context:
This profit and loss question involves two different selling prices for the same item, one resulting in a loss and the other intended to result in a profit. The cost price remains the same in both scenarios, so once we determine the cost price from the loss situation, it can be used directly to find the selling price corresponding to the desired profit. This pattern is very common in aptitude exams and is a good example of how one scenario can be used to derive missing information in another scenario.

Given Data / Assumptions:
- First selling price (SP1) = Rs. 6,500. - At SP1, Savita incurs a loss of 20% on the cost price. - Second selling price (SP2) should give a profit of 20% on the same cost price. - We must find SP2.

Concept / Approach:
In the loss situation, the relationship is SP1 = CP * (1 - loss%), where loss% = 20%. From this we can compute the cost price CP. In the profit situation, the relationship is SP2 = CP * (1 + profit%), again with profit% = 20%. Both percentages are applied to the same cost price, so after calculating CP once, we can directly use it to calculate the second selling price. This strategy neatly links the loss case and the profit case through a single CP value.

Step-by-Step Solution:
In the first case, loss = 20%, so SP1 = CP * (1 - 0.20) = CP * 0.80. Given SP1 = Rs. 6,500, so 6,500 = 0.80 * CP. Thus, CP = 6,500 / 0.80. Compute CP: 6,500 / 0.80 = 6,500 * (10 / 8) = 6,500 * (5 / 4) = 8,125. Therefore, cost price CP = Rs. 8,125. Now, for a profit of 20%, SP2 = CP * (1 + 0.20) = CP * 1.20. So SP2 = 8,125 * 1.20. Compute SP2: 8,125 * 1.20 = 8,125 * (6 / 5) = 9,750. Therefore, she should have sold the item for Rs. 9,750 to earn a profit of 20%.

Verification / Alternative check:
Check loss scenario: 20% of 8,125 = 0.20 * 8,125 = 1,625; SP1 = 8,125 - 1,625 = 6,500, which matches the given selling price with a 20% loss. Check profit scenario: 20% of 8,125 = 1,625; SP2 = 8,125 + 1,625 = 9,750, matching our computed SP2. Both checks confirm that CP and SP2 are consistent with the required conditions.

Why Other Options Are Wrong:
- Rs. 10,375 would correspond to a profit greater than 20% on the cost price. - Rs. 8,125 is the cost price itself, which would imply zero profit or loss, not 20% profit. - Rs. 9,000 gives a profit percentage less than 20%. - “Cannot be determined” is incorrect because enough information is given to uniquely determine CP and then SP2.

Common Pitfalls:
Some students mistakenly average the two selling prices or treat the 20% loss and 20% profit as cancelling each other, which is wrong. Another error is to calculate the profit percentage based on the lower selling price rather than the cost price. Forgetting to multiply properly by the fractional form 6/5 when going from CP to SP2 is also a common arithmetic slip.

Final Answer:
She should have sold the item for Rs. 9,750 to earn a 20% profit.