Problem 34 A quantity, its 1⁄2, and its 1⁄4, added together, become 10. What is the quantity?
Multiply 11⁄21⁄4 so as to get 10.
| \ | 1 | 11⁄21⁄4 |
| 2 | 31⁄2 | |
| \ | 4 | 7 |
| \ | 1⁄7 | [1]1⁄4 |
| 1⁄41⁄28 | 1⁄2 | |
| \ | 1. |
The total is the quantity required, 51⁄21⁄71⁄14.
Proof.
| \ | 1 | 51⁄21⁄71⁄14 |
| \ | 1⁄2 | 21⁄21⁄41⁄141⁄28 |
| \ | 1⁄4 | 11⁄41⁄81⁄281⁄56. |
The whole numbers and simpler fractions (powers of 1⁄2) make a total of 91⁄21⁄8, the remainder is 1⁄41⁄8. The remaining fractions, namely,
1⁄71⁄141⁄281⁄56
applied to 56, are equal to
844221,
making a total of 21, while 1⁄4 and 1⁄8 make 14 and 7, and so also a total of 21. Therefore the result obtained is correct.
The first step is a multiplication of the second kind (Introduction, page 5) with 11⁄21⁄4. taken as multiplicand and 10 as product. After doubling twice, also taking )4 and doubling this twice, the author finds a combination of products exactly equal to 10, and it is not necessary for him to take the second and third steps that belong to this kind of multiplication when the product is.not obtained directly from the partial products. It is easy to see why this multiplication should be so simple; for we have already enough partial products to make any integer up to 14. Compare the notes to Problem 37.
SECTION VII
Problems 35-38. Division of a Hekat
In these problems in the papyrus the questions are put in a curious way: "I have gone a certain number of times into the hekat-measure, certain parts have been added to me, and I return filled. What is it that says this?" It is stated as if the vessel represented as speaking had gone
- ↑ See Introduction, page 5, footnote 2.
