The Compendious Book on Calculation by Completion and Balancing/On mercantile transactions

by Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī

Layout 2

ON MERCANTILE TRANSACTIONS.

You know that all mercantile transactions of people, such as buying and selling, exchange and hire, comprehend always two notions and four numbers, which are stated by the enquirer; namely, measure and price, and quantity and sum. The number which expresses the measure is inversely proportionate to the number which expresses the sum, and the number of the price inversely proportionate to that of the quantity. Three of these four numbers are always known, one is unknown, and this is implied when the person inquiring says how much? and it is the object of the question. The computation in such instances is this, that you try the three given numbers; two of them must necessarily. be inversely proportionate the one to the other. Then you multiply these two proportionate numbers by each other, and you divide the product by the third given number, the proportionate of which is unknown. The quotient of this division is the unknown number, which the inquirer asked for; and it is inversely proportionate to the divisor.^[1]

Examples.— For the first case: If you are told, “ten (49) for six, how much for four?” then ten is the measure; six is the price; the expression how much implies the unknown number of the quantity; and four is the number of the sum. The number of the measure, which is ten, is inversely proportionate to the number of the sum, namely, four. Multiply, therefore, ten by four, that is to say, the two known proportionate numbers by each other; the product is forty. Divide this by the other known number, which is that of the price, namely, six. The quotient is six and two-thirds; it is the unknown number, implied in the words of the question “how much?” it is the quantity, and inversely proportionate to the six, which is the price.

For the second case: Suppose that some one ask this question: “ten for eight, what must be the sum for four?” This is also sometimes expressed thus: “What must be the price of four of them?” Ten is the number of the measure, and is inversely proportionate to the unknown number of the sum, which is involved in the expression how much of the statement. Eight is the number of the price, and this is inversely proportionate to the known number of the quantity, namely, four. Multiply now the two known proportionate numbers one by the other, that is to say, four by eight. The product is thirty-two. Divide this by the other known number, which is that of the measure, namely, ten. The quotient is three and one-fifth; this is the number of the sum, and inversely proportionate to the ten which was the divisor. In this manner all computations in matters of business may be solved.

If somebody says, “a workman receives a pay of ten (50) dirhems per month; how much must be his pay for six days?” Then you know that six days are one-fifth of the month; and that his portion of the dirhems must be proportionate to the portion of the month. You calculate it by observing that one month, or thirty days, is the measure, ten dirhems the price, six days the quantity, and his portion the sum. Multiply the price, that is, ten, by the quantity, which is proportionate to it, namely, six; the product is sixty. Divide this by thirty, which is the known number of the measure. The quotient is two dirhems, and this is the sum. This is the proceeding by which all transactions concerning exchange or measures or weights are settled.

---

1. If $a$ is given for $b$, and $A$ for $B$, then $a:b::A:B$ or $aB=bA\;\therefore \;a={\tfrac {bA}{B}}$ and $b={\tfrac {aB}{A}}$.