Page:The Algebra of Mohammed Ben Musa (1831).djvu/127

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parts. Subtract therefrom the two shares (of a daughter) or one hundred and eighteen parts; there remain eighty parts. Subtract now that which was excepted, namely, one-fourth and one fifth of these eighty, or thirty-six parts; there remain for the legatee eighty-two parts. Deduct this from the parts in the total number of parts in the heritage, namely, four hundred and ninety-five. There remain four hundred and thirteen parts to be distributed into seven shares; the daughter receiving (one share or) fifty-nine (parts), and each son twice as much.

If he leaves two sons and two daughters, and bequeaths to some person as much as the share^[1]. of a

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1. Since there are two sons and two daughters, each son receives ${\tfrac {1}{3}}$, and each daughter ${\tfrac {1}{6}}$ of the residue. Let $v$ = a daughter’s share. Let the 1st legacy $=x=v-{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]$
   Let the 2d legacy $=y=v-{\tfrac {1}{3}}[{\tfrac {1}{3}}-x-v]$
   and 3d legacy $={\tfrac {1}{12}}$
   ${\begin{aligned}&1-{\tfrac {1}{12}}-x-y=6v\\&{\text{i.e.}}\;{\tfrac {2}{3}}-{\tfrac {1}{12}}+{\tfrac {1}{3}}-x-v+{\tfrac {1}{3}}[{\tfrac {1}{3}}-x-v]=6v\\&{\text{or}}\;{\tfrac {2}{3}}-{\tfrac {1}{12}}+{\tfrac {4}{3}}[{\tfrac {1}{3}}-x-v]=6v\\&{\text{i.e.}}\;{\tfrac {7}{12}}+{\tfrac {4}{3}}[{\tfrac {1}{3}}-v+{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]-v]=6v\\&{\text{or}}\;{\tfrac {7}{12}}+{\tfrac {4}{3}}[{\tfrac {6}{5}}[{\tfrac {1}{3}}-v]-v]6v\\&{\text{or}}\;{\tfrac {7}{12}}+{\tfrac {8}{15}}=[6+{\tfrac {4\times 11}{3\times 5}}]v={\tfrac {134}{15}}v\\&{\text{or}}\;{\tfrac {7}{4}}+{\tfrac {8}{5}}={\tfrac {134}{5}}a\;\therefore \;v={\tfrac {67}{536}}={\tfrac {1}{8}}\\\end{aligned}}$
   The 1st Legacy $=x={\tfrac {1}{12}}$
   The 2d Legacy $=y={\tfrac {1}{12}}$
   A son’s share $={\tfrac {1}{4}}$