The Ass and the Mule
Outline Mathematics
Word Problems
Among the many Aesop's Fables there is one which I believe, in time, developed mathematical contents.
A muleteer set forth on a journey, driving before him an Ass and a Mule, both well laden. The Ass, as long as he traveled along the plain, carried his load with ease, but when he began to ascend the steep path of the mountain, felt his load to be more than he could bear. He entreated his companion to relieve him of a small portion, that he might carry home the rest; but the Mule paid no attention to the request. The Ass shortly afterwards fell down dead under his burden. Not knowing what else to do in so wild a region, the Muleteer placed upon the Mule the load carried by the Ass in addition to his own, and at the top of all placed the hide of the Ass, after he had skinned him. the Mule, groaning beneath his heavy burden, said to himself: "I am treated according to my deserts. If I had only been willing to assist the Ass a little in his need, I should not now be bearing, together with his burden, himself as well."
Even as it is, the story is of great moral value and is worth a note. However, I am inclined to think that the conversion between the beasts was more extensive and more detailed. A part of it was lost in antiquity and only surfaced in later mathematical texts. As a matter of fact, the beasts have been loaded with sacks of grain to be delivered to the nearest mill. To the best of my knowledge, the omitted part went like this:
The ass, who was loaded with more sacks than the mule, pleaded with the latter to pick up just one sack, arguing that if it did, the burden would be fairly distributed between the two: each would have the same amount of sacks. The mule who smirked at the suggestion, had callously replied that had the ass picked one sack from his load, he, the mule, would be left with half as many sacks as the ass. How many sacks did each of them carry?
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Copyright © 19962018 Alexander Bogomolny
Solution
The ass and the mule have been loaded with sacks of grain to be delivered to the nearest mill. The ass, who was loaded with more sacks than the mule, pleaded with the latter to pick just one sack, for if it did, the burden would be fairly distributed between the two: each would have the same amount of sacks. The mule who smirked at the suggestion, had callously replied that had the ass picked one sack from his load, he, the mule, would be left with half as many sacks as the ass. How many sacks did each of them carry?
(In the text below, some words are omitted. These have been underlined. Click just above the line. See what happens.)
Assume the ass had A sacks and the mule M sacks. Had a mule picked one of the ass' sack the would have
(1)  A  1 = M + 1,M + 1,M  1,2(M  1),M + 2. 
If a sack was moved the other way round, the mule would get
(2)  A + 1 = 2(M  1),M + 1,M  1,2(M  1),M + 2. 
As usual, there are many ways to solve a system of two linear equations. Here's one possibility. Rewrite the first equation as
(1')  A = M + 2,M + 1,M  1,2(M  1),M + 2, 
which can be substituted into the second equation:
(M + 2) + 1 = 2(M  1),M + 1,M  1,2(M  1),M + 2.
This simplifies to
M + 3 = 2M  2,
and further
2M  M = 3 + 2,3 + 2,3 + 1,3,3  1,3  2,
which immediately yields M = 5 , 4 , 5 , 6 , 7 . From (1'), A = 7 , 4 , 5 , 6 , 7 .
(Do not forget to check your answer.)
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