Page:The Algebra of Mohammed Ben Musa (1831).djvu/129

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and one;^[1] the whole capital will be one thousand six hundred and eight.

If the heirs are the same, and he bequeaths to a person as much as the share of a daughter, and one-fifth of what remains from one-third after the deduction of that share; and to another as much as the share of the second daughter and one-third of what remains from one-fourth after the deduction of that share: then, in the computation,^[2] you must consider that the two legacies are determined by one-fourth and one-third. Take one-third of the capital, and subtract from it one share; there remains one-third of the capital less one share. Then subtract one-fifth of the remainder, namely, one-fifth of one-third of the capital, less one-fifth of the share; there remain four-fifths of one-third, less four-fifths of the share. Then take also one-fourth of the capital, and subtract from it one (84) share; there remains one-fourth of the capital, less one share. Subtract one-third of this remainder: there

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1. ${\tfrac {201}{1608}}={\tfrac {1}{8}}={\tfrac {3}{24}}=v;\;{\text{and}}\;{\tfrac {1}{12}}={\tfrac {2}{24}}=y$
   The common denominator 1608 is unnecessarily great.
2. Let $x$ be the 1st legacy; $y$ the 2d; $v$ a daughter’s share.
   ${\begin{aligned}&1-x-y=6v\\&x=v+3{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]\\&y=v+3{\tfrac {1}{3}}[{\tfrac {1}{4}}-v]\\&{\text{Then}}\;1-{\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{3}}-v-{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]+{\tfrac {1}{4}}-v-{\tfrac {1}{3}}[{\tfrac {1}{4}}-v]=6v\\&{\text{or}}\;{\tfrac {5}{12}}+{\tfrac {4}{5}}[{\tfrac {1}{3}}-v]+{\tfrac {2}{3}}[{\tfrac {1}{4}}-v]=6v\\&\therefore \;{\tfrac {5}{12}}+{\tfrac {4}{15}}+{\tfrac {2}{12}}=[6+{\tfrac {4}{5}}+{\tfrac {2}{3}}]v\\&\therefore \;{\tfrac {51}{60}}={\tfrac {112}{15}}v\;\therefore \;{\tfrac {51}{448}}={\tfrac {153}{1344}}\\&x={\tfrac {212}{1344}}\;;\;y={\tfrac {214}{1344}}\end{aligned}}$

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