Page:The Algebra of Mohammed Ben Musa (1831).djvu/126

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two-fifths of the capital less two (such) shares. Add thereto what was excepted from the legacy, namely, one-fourth and one-fifth of the two-fifths less nine-tenths of (a daughter’s) share.^[1] Then you have two-fifths and nine-tenths of one-fifth of the capital less two (daughter’s) shares and nine-tenths. Add to this three-fifths of the capital. Then you have one capital and nine-tenths of one-fifth of the capital less two (daughter’s) shares and nine-tenths, equal to seven (such) shares. Reduce this by removing the two shares and nine-tenths and adding them to the seven shares. Then you have one capital and nine-tenths of one-fifth of the capital, equal to nine shares of a daughter and nine-tenths. (82) Reduce this to one entire capital, by deducting nine fifty-ninths from what you have. There remains the capital equal to eight such shares and twenty-three fifty-ninths. Assume now each share (of a daughter) to contain fifty-nine parts. Then the whole heritage comprizes four hundred and ninety-five parts. Two-fifths of this are one hundred and ninety-eight

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1. v=${\tfrac {1}{7}}$ of the residue = a daughter’s share.
   $2v$ = a son’s share
   ${\begin{aligned}1-2v+{\tfrac {9}{20}}[{\tfrac {2}{5}}-2v]=7v\\{\text{i.e.}}\;{\tfrac {3}{5}}+{\tfrac {2}{5}}-2v+{\tfrac {9}{20}}[{\tfrac {2}{5}}-2v]=7v\\\therefore \;{\tfrac {3}{5}}+{\tfrac {29}{20}}[{\tfrac {2}{5}}-2v]=7v\\\therefore \;{\tfrac {3}{5}}+{\tfrac {29}{10\times }}=[7+{\tfrac {29}{10}}]v\;\therefore \;{\tfrac {99}{5}}=99v\end{aligned}}$
   $\therefore \;v={\tfrac {59}{495}}$; a son’s share = ${\tfrac {118}{495}}$
   and the legacy to the stranger = ${\tfrac {82}{495}}$