Problem 64
Example of determining an arithmetical progression. Suppose it is said to thee, Distribute 10 hekat of barley among 10 men in such a way that the shares shall he in arithmetical progression with a common difference of 1⁄8, hekat. What is the share of each?
The average share is 1 hekat. The number of differences will be 9, 1 less than the number of men. Take 1⁄2 of the common difference; it makes 1⁄16 hekeat. Multiply this by 9; it makes 1⁄21⁄16 hekat. Add it to the average share; 11⁄21⁄16 hekat is the largest share. Subtract the 1⁄8 hekat for each man until you reach the last.
Do it thus: The ten portions will be 1 1⁄21⁄16, 11⁄41⁄81⁄16, 11⁄41⁄16, 11⁄81⁄16, 1⁄21⁄41⁄81⁄16, 1⁄21⁄41⁄16, 1⁄21⁄81⁄16, 1⁄21⁄16, 1⁄41⁄81⁄16.
The total is 10 hekat.
This problem is explained in the Introduction, page 30. It will be seen from the Literal Translation that the author wrote "Example of distributing the differences," but the common difference is one of the numbers given in the statement of the problem, and the problem is to get the numbers in the progression itself.
Problem 65
Example of dividing 100 loaves among 10 men, including a boatman, a foreman, and a door-keeeper, who receive double portions. What is the share of each?
The working out. Add to the number of the men 3 for those with double portions; it makes 13. Multiply 13 so as to get 100; the result is 72⁄31⁄30. This then is the ration for seven of the men, the boatman, the foreman, and the door-keeper receiving double portions.
For proof we add 72⁄31⁄39 taken 7 times and 151⁄31⁄261⁄78 taken 3 times for the boatman, the foreman, and the door-keeper. The total is 100.
Problem 66
If 10 hekat of fat is given out for a year, what is the amount used in a day?
The working out. Reduce the 10 hekat to ro; it makes 3200. Reduce the year to days; it makes 365. Get 3200 by operating on 365. The
