added together, will give the numerator of the required result, in this case 82, so that the sum of the fractions will be 82⁄105.
Suppose that an Egyptian had the same problem. Since a fractional expression with the Egyptians was a sum of fractions, or of a whole number and fractions, we might think that he could add the two groups by writing down the four fractions together, as 1⁄3 1⁄5 1⁄5 1⁄7, but one law of the Egyptian fraction-system required that the fractions in an expression should all be different. To add the fractions, he would think of a number or of a group of things of some kind, to which he could suppose them to refer, a number or group such that each fraction of it will be a whole number or at least one that he can add to other numbers of the same kind. He might take 105, say 105 loaves, and suppose that he is to receive 1⁄3 1⁄5 of them and also 1⁄5 1⁄7 of them. He would say that ⅓ of 105 loaves is 35 loaves, and 1⁄5 is 21, making 56 loaves in the first group, and that 1⁄5 is 21 loaves and 1⁄7, 15, making 36 loaves in the second group, and that he will receive in all 82 loaves as his part of 105. But he could not say that he will have 82⁄105 of them all. All that he can do is to seek to express the 82 loaves as an aggregate of parts—different parts—of 105 loaves.
To find what parts of 105 loaves will make 82, he would take fractional multipliers and seek to multiply 105 so as to get 82. This would be a multiplication of the second kind (see page 5). He might say[1]
| 1 | 105 | |
| \ | ⅔ | 70 |
| ⅓ | 35 | |
| 1⁄30 | 3½ | |
| \ | 1⁄15 | 7 |
| 1⁄10 | 10½ | |
| 1⁄5 | 21 | |
| \ | 1⁄21 | 5 |
| Total | ⅔ 1⁄15 1⁄21 |
But he might be able without going through a formal multiplication to separate 82 loaves into groups which he recognizes as parts of 105. Thus he might take 70, which is ⅔ of 105, leaving 12, and these he could separate into 7 and 5, which are 1⁄15 and 1⁄21 of 105.
If now he finds in some way that 82 of the loaves will be ⅔ 1⁄15 1⁄21 of the 105 loaves, then he will know that ⅓ 1⁄5 plus 1⁄5 1⁄7 make ⅔ 1⁄15 1⁄21 when applied to 105 loaves, and he will conclude that they always make ⅔ 1⁄15 1⁄21.
- ↑ The Rhind Papyrus contains multiplications for this purpose in Problems 21 and 22.
