STRATEGY CHECK: One option here is to let x = the total number of hats, and then calculate each value in terms of x.
Another option is to assign a "nice" to the total number of hats (i.e., a number that works well with all of the fractions)
Let's take the second approach.

The least common multiple of the three denominators (4, 3 and 5) is 60
So, let's say there are 60 hats in the job lot.

A milliner bought a job lot of (60) hats 1/4 of which were brown.
1/4 of 60 = 15, which means there were 15 brown hats in the job lot.

The milliner sold 2/3 of the hats including 4/5 of the brown hats.
2/3 of 60 = 40, which means the milliner sold a total of 40 hats, and 20 hats remained unsold
4/5 of 15 = 12, which means the milliner sold a total of 12 brown hats, and 3 brown hats remained unsold

What fraction of the unsold hats were brown?
3 of the 20 unsold hats were brown.
Answer = 3/20 = C