This problem requires the use of Expectation. So first let us figure out all the possible scenarios of the game.
First, you pay $c to play the game. The results of the game can be of the following:
Case 1: You draw a jack or a queen. You get $(3-c). In order for case 1 to happen, you must draw a jack or a queen. This implies that the probability of case 1 happening is equal to the probability of you drawing a jack or a queen out of the deck of 52 cards. This probability is equal to (4 jacks + 4 queens)/(52 total cards) = 8/52.
Case 2: You draw a king or an ace. You get $(5-c). In order for this to happen, you must draw either a king or an ace out of the 52 cards. The probability of that happening is also 8/52. (Why?)
Case 3: You didnt draw a queen, jack, king, nor ace. You get $(0-c) in return. The probability of this happening is 36/52. (Why?)
Now we can calculate the expected value of the reward of game:
E[reward of the game] = (3-c)*(8/52) + (5-c)*(8/52) + (0-c)*(16/52)
We've established that in order for this game to be a fair game, the Expected reward must = 0. So therefore set the above eqn to zero and solve for c.
E[reward of the game] = (3-c)*(8/52) + (5-c)*(8/52) + (0-c)*(16/52) = 0