An item is sold at two successive gains of 30% and 20%. If the final selling price after both gains is Rs. 31,200, what is the original cost price of the item?

Introduction / Context:
This problem involves successive gains on the same item. Instead of a single profit percentage, the seller first increases the price by 30% and then further increases the new price by 20%. The final selling price after these two increases is given, and we are asked to determine the original cost price. This type of question helps build understanding of successive percentage changes and reverse calculation from selling price to cost price.

Given Data / Assumptions:

• Let the original cost price be C.
• First gain = 30%, so first selling price = C * 1.30.
• Second gain = 20%, so final selling price = (C * 1.30) * 1.20.
• Final selling price after both gains = Rs. 31,200.
• We need to find the original cost price C.

Concept / Approach:
Successive percentage gains multiply as factors. A gain of 30% corresponds to multiplying the current price by 1.30. A subsequent gain of 20% corresponds to multiplying by 1.20. Together, the combined multiplier is 1.30 * 1.20. We equate this combined factor times the cost price to the known final selling price and solve for the cost price by dividing by the factor. This is a standard reverse-percentage technique.

Step-by-Step Solution:
Let cost price be C.After the first gain of 30%, price becomes C * 1.30.After the second gain of 20%, price becomes C * 1.30 * 1.20.Therefore, final selling price SP = C * 1.30 * 1.20.Compute combined multiplier: 1.30 * 1.20 = 1.56.We are given SP = 31,200, so 1.56 * C = 31,200.Thus C = 31,200 / 1.56.Compute C = Rs. 20,000.Hence, the original cost price of the item is Rs. 20,000.

Verification / Alternative check:
Starting from C = Rs. 20,000, apply 30% gain: 20,000 * 1.30 = Rs. 26,000.Then apply 20% gain on 26,000: 26,000 * 1.20 = Rs. 31,200.This matches the given final selling price, confirming that C = Rs. 20,000 is correct.

Why Other Options Are Wrong:
If C = 15,000, final SP would be 15000 * 1.56 = 23,400, not 31,200.If C = 22,250, final SP would be 22,250 * 1.56, which exceeds 34,000, inconsistent with the given 31,200.If C = 24,000, final SP would be 24,000 * 1.56 = 37,440, also not matching.Only C = 20,000 produces SP = 31,200.

Common Pitfalls:
A common mistake is to add the percentages directly (30% + 20% = 50%) and treat the final price as 150% of C, which is incorrect.Another pitfall is dividing by 1.50 instead of using the correct combined factor of 1.56.Some students also mis-handle decimals and end up with approximate values that do not match any option.

Final Answer:
The original cost price of the item is Rs. 20,000.