If 10% profit of A is equal to 20% loss of B is equal to 15% loss of C, then what is the ratio of the cost price of A, B and C?

The profit and loss difference in this question is a misdirection. 
The profit on A is equal to the loss on B simply means that the exact values of these are equal. Them being a profit or loss would not make any difference. 

we are given, 
$$10\%\ of\ A\ =\ 20\%\ of\ B\ =\ 15\%\ of\ C$$
= $$2A\ = 4B = 3C$$

From this we can get, A:B = 2:1 and B:C = 3:4
giving us, A:B:C to be 6:3:4

Option A is the correct answer.