Two barrels, a and b, contain unspecified amounts of cider, a containing more than b. from a, pour into b as much cider as b already contains. then, from b, pour into a as much cider as a now contains. finally, pour from a into b as much cider as b now has. both barrels now contain 64 liters of cider. how many liters of cider were in each barrel at the start of the process?

You should go backwards to answer this question
There were 3 steps
Step 1 : Pour from A into B
Step 2: Pour from B into A
Step 3 : Pour again from A into B
And in all three cases the amount of cider gets doubled in the container

Now we go Backwards From step 3
B now contains 64L poured from A. That means previously it was half of 64, i.e.,
B contained half of 64 = 32L
A contained (64+32) = 96L
[We are adding 32L because the amount of cider got decreased in A is the amount of cider got increased in B or vice versa]

Now coming to second step doing same thing, but now it is half of A as B poured into A,
A contained half of 96 = 48L
B contained (32+48) = 80L

Now same thing in Step 1,
B contained half of 80 = 40L
A contained (48+40) = 88L

Therefore the amount of cider contained in the containers at the starting of process is,
A = 88L
B = 40L

[If you want to check you answer, you can put the values of 88 and 40 and go stepwise as written in question to check whether it coming correct or not]