that is, the whole of 30, or 1. The answer is1⁄41⁄531⁄1061⁄212
Proof.
| 1 | 1⁄41⁄531⁄1061⁄212 | |
| 2 | 1⁄21⁄301⁄3181⁄7951⁄531⁄106 | |
| 1⁄/3 | 1⁄121⁄1591⁄3181⁄836 | |
| 1⁄/5 | 1⁄201⁄2651⁄5301⁄1060 |
The larger fractions are 1⁄2 and 1⁄4. In order to get 1 we should have for the sum of the remaining fractions %. To get this apply these fractions to 1060.
The fractions
| 1⁄53 | 1⁄106 | 2⁄12 | |||||
| as parts of 1060 make | 20 | 10 | 5 | or | 35 | ||
| 1⁄30 | 1⁄318 | 1⁄795 | 1⁄53 | 1⁄106 | |||
| make | 351⁄3 | 31⁄3 | 11⁄2 | 20 | 10 | " | 70 |
| " | 1⁄12 | 1⁄159 | 1⁄318 | 1⁄636 | |||
| " | 881⁄3 | 62⁄3 | 31⁄3 | 12⁄3 | " | 100 | |
| 1⁄20 | 1⁄265 | 1⁄530 | 1⁄1060 | ||||
| " | 53 | 4 | 2 | 1 | " | 60 |
The total is 265, or 1⁄4 of 1060; for
| 1 | 1060 | |
| 1⁄2 | 530 | |
| 1⁄4 | 265 | |
| 1⁄4 | 265 | |
| Total | 1060. |
In multiplying 1 at the beginning of the solution by 31⁄31⁄5, instead of saying, once 1, twice 2, etc., our author actually writes down "once 1" three times and then the rest of the multiplication, and in getting 1 by operating on 31⁄31⁄5, instead of multiplying this expression directly by multipliers that will eventually give him 1, he applies it to 30, noting that 3 times 30 and 1⁄31⁄5 of 30 make 106. Therefore to find how many times 31⁄31⁄5 will make 1 he determines how many times 106 will make 30 and the answer to this, 1⁄41⁄531⁄1061⁄212, is the answer to the problem.[1]
Problem 37
I have gone three times into the hekat-measure, my 1⁄3 has been added to me, 1⁄3 of my 1⁄3 has been added to me, and my 1⁄9 has been added to me; I return having filled the hekat-measure. What is it that says this?
- ↑ This is the only time that he applies his fractional expressions to a particular number for the purpose of dividing. Generally he uses the method for an addition or subtraction. See Introduction, page 9, footnote.
