Let the player has x hits in y at-bats.
Since players initial average is 0.256, therefore, we can set up an equation:

Further, we are given that player hits 12 in 35 at-bats. Therefore, the total hits are now
and total at-bats are now
.
Since the new average becomes 0.275, therefore, we can set up:

Our next step is to solve these two equations and get values of x and y.

Upon using substitution method, we get:

We can find the value of x by substituting this value of y in the first equation.

Therefore, player made 32 hits in 125 at bats.