(This ancient problem illustrates the origin of "algebra" in INHERITANCE problems, as well as illustrating other matters cited below.)An Arabic man is riding a camel across a desert expanse, when he encounters a novel sight. Three young Arabic men are fiercely arguing, surrounded by 17 camels. Dismounting, the stranger was told the problem. Their father had died, leaving (as their only real inheritance) these 17 camels. Now, the eldest son was to receive half of the camels; the second son, one-third of the camels; the youngest son, one-ninth of the camels. Problem: how could they thus divide the 17 camels?
The stranger adjoined his camel to the collection, making it 18 camels>. Then, the stranger apportioned 9 (= 1/2(18)) camels to the eldest son; 6 (= 1/3(18)) camels to the 2nd son; 2 (= 1/9(18)) camels to the youngest son. Having solved the problem and assuaged their argument, the stranger mounted his own camel and rode away.
This problem shows how "algebra" is DUAL TO ARITHMETIC, working backwards. You know the answer (here, 17), but not how to set up the x so that the arithmetic will work out.
How did it work out? Add the fractions, 1/2 + 1/3 + 1/9. To add them, you must convert each to a fraction with their LEAST COMMON DENOMINATOR (LCD). Well, LCD = 2 · 9 = 18. So, 1/2 = 9/18; 1/3 = 6/18; 1/9 = 2/18. Hence, 1/2 + 1/3 + 1/9 = 9/18 + 6/18 + 2/18 = (9 + 6 + 2)/18 = 17/18. Then, the Algebraic Problem reads: (17/18)x = 17, or (dividing both sides by 17) x/18 = 1, or (multiplying both sides by 18) x = 18.
That is, x MUST BE MADE 18 TO SOLVE THE PROBLEM BY "FRACTIONAL APPORTIONMENT". So, the stranger ADJOINS HIS CAMEL TO THE 17 MAKING THE TOTAL 18 CAMELS, which is a MULTIPLE of 2, 3, 9, hence, easily APPORTIONED.
(Actually, this problem is present in an ancient Egyptian document, the Ahmes Papyrus -- a.k.a. Rhind Mathematical Papyrus. And Leonardo of Pisa (1175?-1250?) -- a.k.a. Fibonacci -- generalzed it.)
This problem also illustrates what Canadian mathematician, Z. A. Melzak, calls "the strategy of adjoining something to solve a problem". For example, ancient Greek geometer Appollonius analyzed properties of 2-D conics by turning this into a 3-D problem, solving it, then going back to 2-D.
This suggests a regular ploy in linear programming algebra, for maximizing or minimizing in diets, business, defense, etc. (Math of this kind was used in American strategy and resulted in two events which changed history!)
This is, of course, a special case of the "conjugation" or BYPASS STRATEGY, which Melzak taught me. (Solving a difficult or "impossible" problem by transforming it into a known, solvable problem; solving; then transforming back into original terms.) The camel problem can be easily diagrammed:
"Whole number" dividing of 17 camels ------------------------------ Adjoin | ^ Remove camel, camel to | | returning change | | to original set | | set V<---------------------------> Apportion camels in new setHere's a TABLE OF BYPASSES in many different fields of activity.