The total is 14 1⁄41⁄561⁄971⁄1941⁄3881⁄6791⁄776, which multiplied by 12⁄31⁄21⁄7 makes 33.
Problem 32
A quantity, its 1⁄3, and its 1⁄4, added together, become 2. What is the quantity?
Multiply 11⁄31⁄4 so as to get 2.
| 1 | 11⁄31⁄4 | |
| \ | 2⁄3 | 11⁄18 |
| \ | 1⁄3 | 1⁄21⁄36 |
| \ | 1⁄6 | 1⁄41⁄72 |
| \ | 1⁄12 | 1⁄81⁄144 |
Take 12 times 12.
| 1 | 12 | |
| 2 | 24 | |
| \ | 4 | 48 |
| \ | 8 | 96 |
| Total | 144. | |
We will apply our fractions to 144. For the given expression we have
| \ | 1 | 144 |
| \ | 1⁄3 | 48 |
| \ | 1⁄4 | 36 |
| Total | 228. | |
The products above, taken as parts of 144, are equal to
228152763819.
The sum of the numbers here that correspond to the multipliers checked is equal to 285 and requires 3 more to make up 288, or 2 times 144. As 11⁄31⁄4 times 144 is 228 we shall have as a continuation of our first multiplication,
| \ | 1⁄228 | 1⁄144 | or | 1 | as | a | part | of | 144 |
| \ | 1⁄114 | 1⁄72 | " | 2 | " | " | " | " | 14 |
Adding together all the multipliers checked in this multiplication, we have 11⁄61⁄121⁄1141⁄228 as the required quantity.
the former multiplication are the multipliers of the latter and so are represented in the left-hand column here, while the multipliers of the former are the alternative numbers, 1, 2, 1⁄2, and 1⁄4, given at the right.
