Why does US currency have a 25 cent piece instead of a 20 cent piece? Why is there a 20 dollar bill instead of a 25 dollar bill? Why aren't both denominations the same, 20 or 25 for both the dollars and cents?
This is because Americans were used to dealing in quarters at the time the denomination was chosen. During the colonial period, a common unit of currency was one eighths of a Spanish real de a ocho. Since each of these Spanish dollars were worth eight Spanish reales it was habitual to divide the eight-real coin into 12.5% wedges known as bits. Two of these were therefore equivalent to 25% of the Spanish dollar, i.e. 25 cents.
When the nascent United States adopted its own currency standard, the original one dollar was based on the aforementioned Spanish dollar. The customary unit of two bits of the Spanish dollar thus became 25 cents of the equivalent United States dollar.
In 1793, Congress adopted decimal coinage - 100 cents to the dollar - to replace the eights into which Spanish dollars were divided. Nevertheless, old habits died hard. Congress also instructed the U.S,. Mint to coin quarter dollars. Quarters made no sense decimally, but they reflected the partition of a Spanish dollar into eight reales.
- Conlin, Joseph. The American Past: A Survey of American History, Volume II: Since 1865. Cengage Learning, 2013.
I originally thought it was because of best way to reduce number of coins needed before I read Semaphore's answer.
I thought it was designed this way due to efficiency: for example at all values between 0 and 1 dollar that ends in 0 or 5
5 cent = 1 coin
10 cent = 1 coin
15 cent = 2 coins
20 cent = 2 coins (1 coin if using 5/10/20)
25 cent = 1 coin (2 coins if using 5/10/20)
30 cent = 2 coins
35 cent = 2 coins (3 coins if using 5/10/20)
40 cent = 3 coins (2 coins if using 5/10/20)
45 cent = 3 coins
50 cent = 2 coins (3 coins if using 5/10/20)
55 cent = 3 coins (4 coins if using 5/10/20)
60 cent = 3 coins
65 cent = 4 coins
70 cent = 4 coins
75 cent = 3 coins (5 coins if using 5/10/20)
80 cent = 4 coins
85 cent = 4 coins (5 coins if using 5/10/20)
90 cent = 5 coins
95 cent = 5 coins (6 coins if using 5/10/20)
1 dollar = 4 coins (5 coins if using 5/10/20)
There are only 2 occasions that 5/10/20 uses less coins than 5/10/25
I think the real answer basically boils down to attrition. Over the years, the US has minted many denominations of coins, ranging from the half cent piece up to the $50 Half-Union coin: https://en.wikipedia.org/wiki/Coins_of_the_United_States_dollar#Obsolete_coins
Most of them have fallen out of use for various reasons, mostly because having many denominations proved inconvenient and/or confusing. The US did mint a 20 cent piece for a few years, but it was unpopular because it was easily confused with a quarter: https://en.wikipedia.org/wiki/Twenty-cent_piece_(United_States_coin) The same was true of the short-lived Susan B. Anthony dollar coin https://en.wikipedia.org/wiki/Dollar_coin_(United_States)#Susan_B._Anthony_dollar_(1979%E2%80%931981;_1999) which I suspect (along with the inconvenience of having two $1 coins of very different sizes) contributed to the current unpopularity of the dollar coin. (Gold coins are obviously a special case, thanks to FDR &c.)
I'd guess that the use of cash registers also played a part, as they can easily fit only so many pockets for coins. Perhaps that also caused the virtual abandonment of the $2 bill, and is contributing to the decline I've seen in the use of the $10 bill. (My grocery store only gives out $5 bills in change, and any $10s you give them are tucked underneath the tray.)