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This joke is a classic example of wordplay, cleverly using the double meaning of the word "green." The humor arises from taking a literal physical characteristic of the dollar bill – its color – and combining it with an idiomatic expression that describes a feeling of sickness. It's an unexpected twist that anthropomorphizes the money, giving it a very human ailment for a silly, laugh-out-loud effect.
The reason dollar bills are green dates back to the American Civil War when the U.S. government issued paper currency known as "greenbacks" to finance the war effort. This nickname stuck because the reverse side of these notes was printed with green ink. The choice of green ink was partly practical, as it was readily available, durable, and considered effective at preventing counterfeiting, especially against black-and-white photography of the era. Meanwhile, the idiom "feeling green" or "green around the gills" means to look ill, especially nauseated. The association of green with a sickly complexion has roots dating back centuries, possibly to the 14th century.
So, the joke playfully imagines a dollar bill experiencing literal symptoms of an illness based on its most prominent visual feature and a common English phrase. It’s a lighthearted linguistic trick that relies on our familiarity with both the appearance of currency and the colorful ways we describe feeling under the weather.