Karat, fineness and percentage are one number
Karat is a fraction of 24: 18K means eighteen parts gold in twenty-four, so 75% gold, so 750 parts per thousand. Millesimal fineness writes the same fraction as parts per thousand and is what most modern jewellery is actually stamped with. Percentage is the same again with the decimal moved.
Where the stamp and the arithmetic disagree, the stamp wins, because the stamp is what is on the metal in your hand. 14K works out at 583.3 exactly and is stamped 583 in the US or 585 in Europe. 22K is 916.6 and is stamped 916 or 917. The difference is a fraction of a percent and it is not an error.
What the alloy is for
Pure gold is too soft to hold a stone or survive daily wear, which is why almost no jewellery is 24K. The other metals in the alloy provide hardness and set the colour: copper pushes it red, silver and palladium push it pale, and nickel or palladium with rhodium plating give white gold. A 9K piece is 62.5% other metals, which is why it is markedly harder and paler than 18K.
National floors differ. The US will not let anything under 10K be described as gold. The UK and Ireland allow 9K. Across the Gulf and North Africa 21K is the norm, and in India, Pakistan and Bangladesh 22K dominates, because gold there is bought as a store of value as much as a design — and higher purity means less alloy to lose when it is eventually sold.