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Problem 8
Due in DSC 203 by 3 PM, Friday, October 19, 2001
An old Arab left his camel herd to his 3 sons provided that they split it according to his will which among other things prohibited them from selling any of the camels and splitting the money. The will stipulated that 1/2 of the herd should go to the oldest brother, the second oldest should get 1/3 of it while the youngest should receive 1/9. Since the parental herd numbered exactly 17 camels and 17 is not a multiple of 2,3 or 9 the three brothers started a long dispute about who's getting what. Finally they agreed to ask the advice of the local judge who lived in the neighboring oasis and was known as a very honest and wise man. After hearing their story the judge asked for a one day time-out for meditation but promised to come over the next day and solve the dilema. On the next day he kept his word and appeared riding his own camel. He asked the brothers if they would agree to add his camel to the herd and calculate the fractions out of 18 rather than 17. This way each of them would get more and comply with the stipulations of the will. On their very prompt approval he required that if any remainder existed after splitting the herd this way, then that would be his property for doing the job. As they readily agreed to that as well, the computation began:
1/2 of 18 equals 9, so 9 camels to the oldest.
1/3 of 18 equals 6, so 6 to the second brother.
Finally, 1/9 of 18 is 2, so only 2 camels to the youngest.
Total 9+6+2=17 so remainder 1 camel. After this computation the old judge got on the back of his own camel which accidentally or intentionally was the remainder and off he was, leaving everybody totally bewildered. Explain why this computation was possible, (i.e. explain why he could give more to each of the brothers, comply with the will, and recuperate his property on top of everything). Use calculus to prove that this way of splitting the 17 camels is fair according to the will, if one wants to be consistent with the principle 1/2 to the oldest, 1/3 to the second oldest, and 1/9 to the youngest.
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