Let’s Play Ball!

In baseball and softball, a hit means the batter makes contact with the ball and makes it safely to a base. A player’s batting average represents the ratio of the number of hits over the number of at-bats. However, when a player reaches base without getting a hit or an out, it is not included as an at-bat in the player’s average. For example, a player who goes to the plate 120 times, has 30 hits, 19 walks, and 1 hit by pitch, would have 100 at-bats (100=120-19-1) and a batting average of 30/100 = 0.300 because it represents the player’s ability to get a hit after putting the ball in play.

At the end of the season, a player has an average of exactly 0.336, and had exactly one walk for every three hits. Assuming the player was never hit by a pitch, how many plate appearances did the player have (including walks)?