A man buys 3 type I cakes and 6 type II cakes together for Rs 900. He sells each type I cake at a profit of 15% and each type II cake at a loss of 10%. If his overall profit on the transaction is Rs 30, what are the cost prices (in Rs) of one type I cake and one type II cake?

Introduction / Context:
This question combines linear equations with profit and loss concepts. The man buys two different types of cakes at different unknown costs, then sells them with different profit and loss percentages, and we are told his overall profit amount. We must form equations to find the individual cost prices of the two types of cakes.

Given Data / Assumptions:
- Number of type I cakes bought = 3. - Number of type II cakes bought = 6. - Total cost for all cakes = Rs 900. - Type I cakes are sold at 15% profit. - Type II cakes are sold at 10% loss. - Overall profit on the entire transaction = Rs 30.

Concept / Approach:
Let x be the cost price of one type I cake and y be the cost price of one type II cake. We create two equations: one from the total cost, and another from the total selling price equaling total cost plus profit. We then solve this system of linear equations to find x and y.

Step-by-Step Solution:
Step 1: Total cost equation: 3x + 6y = 900. Step 2: Selling price of each type I cake = x * 1.15. Step 3: Selling price of each type II cake = y * 0.90. Step 4: Total selling price = 3 * 1.15x + 6 * 0.90y = 3.45x + 5.4y. Step 5: Overall profit is Rs 30, so total selling price = 900 + 30 = 930. Step 6: Second equation: 3.45x + 5.4y = 930. Step 7: Solving the system 3x + 6y = 900 and 3.45x + 5.4y = 930 gives x = 160 and y = 70.

Verification / Alternative check:
Compute total cost and selling price with x = 160 and y = 70. Total cost = 3 * 160 + 6 * 70 = 480 + 420 = 900, as required. Selling price of type I cakes = 3 * 160 * 1.15 = 3 * 184 = 552. Selling price of type II cakes = 6 * 70 * 0.90 = 6 * 63 = 378. Total selling price = 552 + 378 = 930. Profit = 930 - 900 = 30, which matches the given profit.

Why Other Options Are Wrong:
Other pairs such as 100,100 or 180,60 do not satisfy both the cost and profit conditions when checked. For example, if both types cost 100, total cost would be 900 but the resulting overall profit would not be 30 under the given percentage gains and losses. Only 160 and 70 satisfy both equations exactly.

Common Pitfalls:
A common error is to apply the profit and loss percentages to the total cost directly instead of separately for each type. Another mistake is to round off intermediate decimals too early, which can lead to incorrect solutions. Keeping the equations symbolic until the final step yields a precise result.

Final Answer:
The cost prices are Rs 160 for each type I cake and Rs 70 for each type II cake.