All of these solutions have the sale of chickens for $0, which makes the problem easy (eg, they each sold 7 chickens for $5 and gave the rest away), but is surely not in the spirit of the question.
For example, one dictionary defines "sell" as "to exchange (something) for money". $0 is not money. It's literally nothing.
I don't know why people (not you) come up with "clever" solutions that violate the plain wording of the question.
After all, part of the linked article's point was how it struck him as potentially broken; if the problem seems broken, then many will heuristically start attacking the problem itself rather than the math. And once the problem's no longer trusted to be stated as exact and correct, well, a simple pedantic jump in meaning is a pretty small step. (I'll concede it's possible you've never come across a broken 'clever' math word problem before, but having found enough to be wary myself - both in school and college - I'm rarely surprised at people leaping for the lateral thinking solution.)
user> (run* [q] (fresh [morning afternoon a b c d e f] (fd/in morning afternoon (fd/interval 0 3500)) (fd/in a b c d e f (fd/interval 0 26)) (fd/eq (< afternoon morning) (> morning 0) (> afternoon 0)
(= (+ a b) 10) (= (+ c d) 16) (= (+ e f) 26) (= (+ (* morning a) (* afternoon b)) 3500) (= (+ (* morning c) (* afternoon d)) 3500) (= (+ (* morning e) (* afternoon f)) 3500)) (== q [morning afternoon a b c d e f])))
Spoiler warning, scroll right: ([375 125 9 1 6 10 1 25])