the parts are thirteen.[1] Take the capital, and subtract from it one-fifth less two parts; and again, subtract one-sixth of the remainder. You retain two-thirds of the capital, which, when increased by one part and two-thirds, are equal to thirteen parts. Subtract the one part and two-thirds from the thirteen parts: there remain two thirds of the capital, equal to eleven parts and one-third. Complete your capital, by adding to the parts as much as their moiety; thus you find the capital equal to seventeen parts. Assume now the capital to be eighty-five, and each part five; then the first legacy is seven, and the second thirteen, and the remaining sixty-five are for the heirs.
If the case is the same, and she bequeaths to some person as much as must be added to the share of the mother, to make it one-third of the capital, less that sum which must be added to make the share of a daughter equal, to one-fourth of what remains of the capital after the deduction of the above complement; then the parts are thirteen.[2] Take the capital, and (96)
![{\displaystyle {\begin{aligned}&1-x-y=13v\\&x={\tfrac {1}{5}}-2v\;;y={\tfrac {1}{6}}[1-x]\\&1-x-{\tfrac {1}{6}}[1-x]=13v\\&\therefore \;{\tfrac {5}{6}}[1-x]=13v\\&\therefore \;{\tfrac {5}{6}}[{\tfrac {4}{5}}+2v]=13v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {5}{3}}v=13v\\&\therefore \;{\tfrac {2}{3}}={\tfrac {34}{3}}v\;\therefore \;v={\tfrac {1}{17}}\;;x={\tfrac {7}{85}}\;;y={\tfrac {13}{85}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/728ba7b770e7b6cb7e0a124db72b5847baab47f0)
![{\displaystyle {\begin{aligned}&1-x+y=13v\;;\;{\text{and}}\;x={\tfrac {1}{3}}-2v\;;y={\tfrac {1}{4}}[1-x]-v\\&\therefore \;1-x+{\tfrac {1}{4}}[1-x]-v=13v\\&\therefore \;{\tfrac {5}{4}}[1-x]=14v\;\therefore \;{\tfrac {5}{4}}[{\tfrac {2}{3}}+2v]=14v\\&\therefore \;{\tfrac {5}{6}}={\tfrac {23}{2}}v\;\therefore \;v={\tfrac {5}{69}}\;;x-y={\tfrac {4}{69}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c517a4ca46bf4379dcff21b712a08042743cba9)
