The rule of transactions (rule of three)

The rule of transactions is a short chapter in al-Khwarizmi's Book of al-Jabr and al-Muqabala, stating that all people's dealings, buying, selling, exchange, hire and the like, rest on four numbers spoken by the questioner: the rated quantity (musa''ar), the rate (si'r), the price (thaman) and the priced quantity (muthamman). ¹

The rated quantity is "opposed" to the price and the rate to the priced quantity; three of the four numbers are always known and one is unknown, the one asked about with "how much". ¹ Multiply the two known opposed numbers together and divide the product by the third known number to obtain the unknown. ¹

• "Ten for six: how much do you get for four?" Ten is the rated quantity, six the rate and four the price; ten times four is forty, divided by six gives six and two-thirds, the priced quantity. ²
• "Ten for eight: what is the price of four?" Four times eight is thirty-two, divided by ten gives three and a fifth. ²
• A hired man paid ten dirhams a month works six days: the month of thirty days is the rated quantity, ten the rate and six days the priced quantity; ten times six divided by thirty gives two dirhams. ²

Al-Khwarizmi says this is how people deal with one another in exchange, measures and weights. ² In the manuscript's margin two lines of verse set the rule out for goods measured or weighed; the editors of the Cairo edition quote them. ¹