Page:The Algebra of Mohammed Ben Musa (1831).djvu/146

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subtract from it one-third less two parts, and add to the remainder one-fourth (of such remainder) less one part; then you have five-sixths of the capital and one part and a half, equal to thirteen parts. Subtract one part and a half from thirteen parts. There remain eleven parts and a half, equal to five-sixths of the capital. Complete the capital, by adding to the parts as much as one-fifth of them. Thus you find the capital equal to thirteen parts and four-fifths. Assume, now, a part to be five, then the capital is sixty-nine, and the legacy four.

“A man dies, and leaves a son and five daughters, and bequeaths to some person as much as must be added to the share of the son to complete one-fifth and one-sixth, less one-fourth of what remains of one-third after the subtraction of the complement.”^[1] Take one-third of the capital, and subtract from it one-fifth and one-sixth of the capital, less two (seventh) parts; so that you retain two parts less four one hundred and twentieths of the capital. Then add it to the exception, which is half a part less one one hundred and

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1. Since there are five daughters and one son, each daughter receives ${\tfrac {1}{7}}$, and the son ${\tfrac {2}{7}}$ of the residue.
   ${\begin{aligned}&1-x=7v\;;{\tfrac {1}{5}}+{\tfrac {1}{6}}={\tfrac {11}{30}}\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{3}}-{\tfrac {11}{30}}+2v+{\tfrac {1}{4}}[{\tfrac {1}{3}}-{\tfrac {11}{30}}+2v]=7v\\&\therefore \;{\tfrac {2}{3}}-{\tfrac {1}{30}}+2v+{\tfrac {1}{4}}[{\tfrac {-1}{30}}+2v]=7v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {5}{4}}[{\tfrac {-1}{30}}+2v]=7v\\&\therefore \;{\tfrac {4}{6}}-{\tfrac {1}{24}}={\tfrac {9}{2}}v\\&\therefore \;{\tfrac {5}{8}}={\tfrac {9}{2}}v\;\therefore \;v={\tfrac {5}{36}},\;{\text{and}}\;x={\tfrac {1}{36}}\end{aligned}}$