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you constitute the parts of the residue by taking them out of thirteen.[1] Take the capital, and subtract from it one-fourth less two parts; and again, subtract one-fifth of what you retain of the capital, less one part; then look how much remains of the capital after the deduction of the parts. This remainder, namely, three-fifths of the capital, when increased by two parts and three-fifths, will be equal to thirteen parts. Subtract two parts and three-fifths from thirteen parts, there remain ten parts and two-fifths, equal to three-fifths of capital. Complete the capital, by adding to the parts which you have, as much as two-thirds of the same. Then you have a capital equal to seventeen parts and one-third. Assume a part to be three, then the capital is fifty-two, each part three; the first legacy will be seven, and the second six.
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-fifth of the capital, and to another one-sixth of the remainder of the capital; then
![{\displaystyle {\begin{aligned}&1-x-y=13v\\&x={\tfrac {1}{4}}-2v\;;y={\tfrac {1}{5}}[1-x]-v\\&1-x-{\tfrac {1}{5}}[1-x]+v=13v\\&{\tfrac {4}{5}}[1-x]=12v\;\therefore \;{\tfrac {4}{5}}[{\tfrac {3}{4}}+2v]=12v\\&\therefore \;{\tfrac {3}{5}}=[12-{\tfrac {8}{5}}]v={\tfrac {52}{5}}v\\&\therefore \;v={\tfrac {3}{52}}\;\therefore \;x={\tfrac {7}{52}},y={\tfrac {6}{52}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf01cac9d2c55e587562bfb193588b935add4e9)
