Page:The Algebra of Mohammed Ben Musa (1831).djvu/138

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If he leaves three sons and two daughters, and bequeaths to some person as much as the share of a daughter plus one dirhem; and to another one-fifth of what remains from one-fourth after the deduction of the first legacy, plus one dirhem; and to a third person one-fourth of what remains from one-third after the deduction of all this, plus one dirhem; and to a fourth person one-eighth of the whole capital, requiring all the legacies to be paid off by the heirs generally: then you calculate this by exhibiting the dirhems distinctly, which is better in such a case.^[1] Take one-fourth of the capital, and assume it to be six dirhems; the entire capital will be twenty-four dirhems. Subtract one share from the one-fourth; there remain six dirhems less one share. Subtract also one dirhem; there remain five dirhems less one share. Subtract

↑ Let the legacies to the three first legatees be, severally, $x,y,z$ ; the fourth legacy = ${\tfrac {1}{8}}$ ; and let a daughters’ share $v$
${\begin{aligned}&\therefore \;{\tfrac {7}{8}}-x-y-z=8v\\&x=v+d;y={\tfrac {1}{5}}[{\tfrac {1}{4}}-x]+d;z={\tfrac {1}{4}}[{\tfrac {1}{3}}-x-y]+d\\&{\text{Then}}\;{\tfrac {7}{8}}-{\tfrac {1}{3}}+{\tfrac {1}{3}}-x-y-{\tfrac {1}{4}}[{\tfrac {1}{3}}-x-y]-d=8v\\&\therefore \;{\tfrac {13}{24}}+{\tfrac {3}{4}}[{\tfrac {1}{3}}-x-y]-d=8v\\&{\text{but}}\;{\tfrac {1}{3}}-x-y={\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{4}}-x-{\tfrac {1}{5}}[{\tfrac {1}{4}}-x]-d\\&={\tfrac {1}{12}}+{\tfrac {4}{5}}[{\tfrac {1}{4}}-x]-d\\&={\tfrac {1}{12}}+{\tfrac {1}{5}}-{\tfrac {4}{5}}v-{\tfrac {9}{5}}d\\&={\tfrac {17}{60}}+{\tfrac {4}{5}}v-{\tfrac {9}{5}}d\\&\therefore \;{\tfrac {13}{24}}+{\tfrac {3}{4}}\times {\tfrac {17}{60}}-{\tfrac {3}{5}}v-[{\tfrac {3}{4}}\times {\tfrac {9}{5}}+1]d=8v\\&\therefore \;{\tfrac {181}{240}}-{\tfrac {47}{20}}d={\tfrac {43}{5}}v\;\therefore \;v={\tfrac {181}{2064}}-{\tfrac {564}{2064}}d,\;{\text{and}}\;1={\tfrac {2064}{181}}v+{\tfrac {564}{181}}d\\&x={\tfrac {181}{2064}}+{\tfrac {1500}{2064}}d;y={\tfrac {67}{2064}}+{\tfrac {1764}{2064}}d;z={\tfrac {110}{2064}}+{\tfrac {1248}{2064}}d\end{aligned}}$

[1] Let the legacies to the three first legatees be, severally, $x,y,z$ ; the fourth legacy = ${\tfrac {1}{8}}$ ; and let a daughters’ share $v$
${\begin{aligned}&\therefore \;{\tfrac {7}{8}}-x-y-z=8v\\&x=v+d;y={\tfrac {1}{5}}[{\tfrac {1}{4}}-x]+d;z={\tfrac {1}{4}}[{\tfrac {1}{3}}-x-y]+d\\&{\text{Then}}\;{\tfrac {7}{8}}-{\tfrac {1}{3}}+{\tfrac {1}{3}}-x-y-{\tfrac {1}{4}}[{\tfrac {1}{3}}-x-y]-d=8v\\&\therefore \;{\tfrac {13}{24}}+{\tfrac {3}{4}}[{\tfrac {1}{3}}-x-y]-d=8v\\&{\text{but}}\;{\tfrac {1}{3}}-x-y={\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{4}}-x-{\tfrac {1}{5}}[{\tfrac {1}{4}}-x]-d\\&={\tfrac {1}{12}}+{\tfrac {4}{5}}[{\tfrac {1}{4}}-x]-d\\&={\tfrac {1}{12}}+{\tfrac {1}{5}}-{\tfrac {4}{5}}v-{\tfrac {9}{5}}d\\&={\tfrac {17}{60}}+{\tfrac {4}{5}}v-{\tfrac {9}{5}}d\\&\therefore \;{\tfrac {13}{24}}+{\tfrac {3}{4}}\times {\tfrac {17}{60}}-{\tfrac {3}{5}}v-[{\tfrac {3}{4}}\times {\tfrac {9}{5}}+1]d=8v\\&\therefore \;{\tfrac {181}{240}}-{\tfrac {47}{20}}d={\tfrac {43}{5}}v\;\therefore \;v={\tfrac {181}{2064}}-{\tfrac {564}{2064}}d,\;{\text{and}}\;1={\tfrac {2064}{181}}v+{\tfrac {564}{181}}d\\&x={\tfrac {181}{2064}}+{\tfrac {1500}{2064}}d;y={\tfrac {67}{2064}}+{\tfrac {1764}{2064}}d;z={\tfrac {110}{2064}}+{\tfrac {1248}{2064}}d\end{aligned}}$

[1]

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