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Nearly all countries have both paper money and coins. Coins are used for low-denomination amounts because they get a lot of use and paper bills would wear out too quickly; they're also easier to use in vending machines and to process in counting machines.

Bills are used for higher amounts because coins would be too large and/or it would be too difficult to create enough easily-distinguished sizes and designs to cover all denominations from 1¢ to $100. Because higher denominations tend to get less wear(*), it's more cost-effective to issue them as bills than coins.

(*) The US is unusual among industrialized countries in its reliance on a relatively low-denomination paper $1 bill. Due to inflation the purchasing power of a $1 bill is less than a quarter from the 1970s and a dime from the 1950s. Thus the $1 bill fills the same role as those coins once did in change-making, and must be produced in enormous quantities each year because they wear out in 18 to 24 months. By contrast most other countries now have coins in amounts equivalent to anywhere from US$2 to US$4, with expected life spans of 30 to 50 years in average use.

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Q: Why does the US have paper money and coins?
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