SECTION VIII
Problems 39-40. Division of Loaves. Arithmetical Progression
Problem 39
Example of finding the difference of share when 100 loaves are divided among 10 men, 50 in equal shares among 6 men and 50 in equal shares among 4. What is the difference of the shares?
Multiply 4 so as to get 50.
| 1 | 4 | |
| \ | 10 | 40 |
| \ | 2 | 8 |
| \ | 1⁄2 | 2 |
| Total | 12 | 1⁄2 |
Multiply 6 so as to get 50.
| 1 | 6 | |
| 2 | 12 | |
| 4 | 24 | |
| \ | 8 | 48 |
| \ | 1⁄3 | 2 |
| Total | 81⁄3. |
Therefore each of the 4 will receive 121⁄2 and each of the 6will receive 81⁄3, and the difference of share will be 41⁄6.
In this problem the result obtained and the methods used are so simple that the problem seems hardly worth while. It is possible that the author intended to state a problem in arithmetical progression like the next one.
We may notice that he considers the 4 first in the solution, although he puts the 6 first in the statement. In the papyrus he expressed the result by writing 121⁄2 four times and 81⁄3 six times.
Problem 40
Divide 100 loaves among 5 men in such a way that the shares received shall he in arithmetical progression and that 1⁄7 of the sum of the largest three shares shall he equal to the sum of the smallest two. What is the difference of the shares?
Do it thus: Make the diflerence of the shares 51⁄2. Then the amounts that the 5 men receive will be
23171⁄21261⁄21, total 60.
