Sharing dromedaries

Feeling his end was near , an old sheik indicated his will to his wise counselor . He wanted to share his livestock as well : half for his eldest son, the third in the second and finally the ninth cadet. Unfortunately his death , his flock consisted of 17 camels ... Then the wise counselor to the neighbor borrowed a camel .. So he had 18 cattle , and he shared : half or 9 senior, third or 6 for the second and finally the ninth or 2 for the cadet. And as 9 + 6 + 2 = 17 ... he gave the 18th animal to its owner. And each of the heirs had the satisfaction of receiving more than his father had given them . The first animal received 1/2 (9 instead of 8.5 ), the second 1/3 of dromedary more and more in the last 1/9

Paradoxical ? But the old counselor was very wise ,
he did not commit injustice ! To you to prove it ...

Solutions : 

Of course the old counselor did not give half to the eldest of the herd or third in the second , nor the third to ninth . This curious result paradoxical at first, can be explained if we notice the sum of the fractions 1/2 , 1/3 and 1/9 of 17/18 is not the unit 18/18 . As a result , following to the letter the instructions of the father and assuming we have been able to share , it would have remained a part of the estate, that is to say 1/18 of this succession without owner. But he respected the proportions between the three heirs. The numbers obtained are proportional to 1/2, 1/3 and finally 1/9 : indeed 6 represents 2/3 of 9 (which is the ratio of 1/3 and 1/2) ; 2 represents 2/9 of 9 (which is the ratio of 1/9 and 1/2) . The sharing is fair .