BİRİNCİ FEN · BİRİNCİ BÂB · ÜÇÜNCÜ FASIL · Onuncu Madde
The Tenth Article Describes the easiest way to find an unknown number. O esteemed one! Know that arithmetic scholars have said: To discover the unknown number, the rule of proportional quadruples is a comprehensive and sound, useful and appropriate method; its benefit is abundant, error-free, accurate, and it is the essence of this operation. For all problems with unknowns can be solved using the rule of proportional quadruples.
A proportional quadruple is four numbers such that the ratio of the first to the second is equal to the ratio of the third to the fourth. In this proportion, the product of the “inward” terms must equal the product of the “outward” terms. If one of the “inward” terms is unknown, you divide the product of the two “outward” terms by the known “inward” term. The result is the “inward” term which is the unknown number. If one of the two “outward” terms is unknown, you divide the product of the two “inward” terms by the known “outward” term. The result is the “outward” term which is the unknown number.
For example, let our proportional numbers be 2, 4, 3, and 6. Here, the ratio of 2 to 4 is equal to the ratio of 3 to 6. The product of the “inward” terms (2 times 6) equals the product of the “outward” terms (3 times 4). If one of these four numbers were unknown, the unknown number could be found from the other three numbers.
If the unknown number is one of the “inward” terms, say 2, you multiply the “outward” terms, 3 and 4; you obtain 12, and divide it by the known “inward” term, 6; the unknown number, 2, is found. If the “inward” term, 6, were unknown, you would first divide the resulting product by the known “inward” term, 2; you would obtain the unknown number, 6. If one of the “outward” terms, say 4, were unknown, you would multiply the two “inward” terms, 2 and 6, to get a result of 12, and divide it by the known “outward” term, 3; this result gives the unknown number, 4. If one of the “outward” terms, say 3, were unknown, you would divide the result of the previous multiplication by the known “outward” term, 4; this result gives the unknown number, 3.
This is the method of proportional quadruples using the operation of multiplication. The method of proportional quadruples using division is as follows: If one of the two “outward” terms is unknown, you divide one of the two “inward” terms by the known “outward” term and multiply the resulting quotient by the other “inward” term. This result gives the unknown “outward” term. If one of the two “inward” terms is unknown, you divide one of the two “outward” terms by the known “inward” term and multiply the resulting quotient by the other “outward” term. This result is the unknown “inward” term.
For example: The ratio of 2 to 3 is equal to the ratio of 6 to 9. If one of the “outward” terms, say 6, were unknown, you would divide one of the “inward” terms, say 9, by the known “outward” term, 3, and multiply the resulting quotient by the other “inward” term, 2. The result gives the unknown “outward” term, 6. If one of the “inward” terms, say 9, were unknown, you would divide one of the “outward” terms, say 6, by the known “inward” term, 2, and multiply the resulting quotient by the other “outward” term, 3. The result gives the unknown number, 9.
If the (inward) unknown number were 2, since the divided is less than the divisor, you would take the fractional number (3 divided by 9) as the division of 3 by 9. You would multiply the resulting quotient, 1/3, by 6. The result gives the unknown number, 2. If (the outward) 3 were unknown, you would divide 9 by 6 and multiply the resulting quotient, 1 and 1/2, by 2. The result gives the unknown number, 3.
Reminder: This operation can be used in addition, subtraction, multiplication, or trade operations.
Example of an Addition Operation: If the question is “By how much must a number be increased to become 3?” the answer to this operation is as follows: The denominator of the fraction is 4. This number is taken as the basis and it is called “me’haz.” Operations are performed on it. That is, if addition is stated, you add; if subtraction is stated, you subtract. The resulting number is called “vasıta.” In this case, there are three knowns: one is the “me’haz,” one is the “vasıta,” and one is the number given by the questioner, which is also called “mâlum.” Because the ratio of the “me’haz” to the “vasıta” is equal to the ratio of the unknown to the fourth, “mâlum.” Therefore, you multiply the “me’haz” by the “mâlum,” obtain a result, and divide it by the “vasıta.” The resulting quotient gives the number we are looking for, 2 and 2/5.
Example of a Subtraction Operation: If the question is “From which number must 1/3 be subtracted to get 6?” the answer to this operation is as follows: To find the answer to this question, you use the method explained above. There are three knowns: one is the “me’haz,” 3; one is the “vasıta,” 2; and one is the “mâlum,” 6. The ratio of 3 to 2 is equal to the ratio of the unknown to 6. You divide the product of the “inward” terms, 18, by the “vasıta,” which is 2. The result gives the unknown number, 9.
Regarding Trade Operations: If 5 bushels cost 3 dirhams, what is the price of 2 bushels? 5 bushels are “müse‘‘ar,” 3 dirhams are “si‘r,” 2 bushels are “müsemmen,” and the desired amount is “semen.” The ratio of the “müse‘‘ar” to the “si‘r” is equal to the ratio of the “müsemmen” to the “semen.” Therefore, the unknown is the fourth term (semen). You divide the product of the two “outward” terms, 6, by the first term (“müse‘‘ar”); the result will be found. If the question were “What is the price of 2 bushels?” and the response were “How many dirhams do 2 bushels cost?” you would divide the product of the two “inward” terms, 10, by the second term (si’r), which is 3. This way, the unknown can be found. In the second question as well, a similar process is followed involving multiplication with…