Page:The Algebra of Mohammed Ben Musa (1831).djvu/112

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of the sons less the amount of the share of the widow.” Divide the heritage into thirty-two parts. The widow receives one-eighth,^[1] namely, four; and each son seven. Consequently the legatee must receive three-sevenths of the share of a son. Add, therefore, to the heritage three-sevenths of the share of a son, that is to say, three parts, which is the amount of the legacy. This gives thirty-five, from which the legatee receives three; and the remaining thirty-two are distributed among the heirs proportionably to their legal shares.

If he leaves two sons and a daughter,^[2] and bequeaths to some one as much as would be the share of a third son, if he had one; then you must consider, what (72) would be the share of each son, in case he had three: Assume this to be seven, and for the entire heritage

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1. A widow is entitled to $\scriptstyle {\frac {1}{8}}$th of the residue; therefore $\scriptstyle {\frac {7}{8}}$ths of the residue are to be distributed among the sons of the testator. Let $x$ be the stranger’s legacy. The widow’s share = ${\tfrac {1-x}{8}}$; each son’s share =${\tfrac {1}{4}}\times {\tfrac {7}{8}}[1-x]$; and a son’s share, minus the widow’s share = $[{\tfrac {7}{4}}-1]{\tfrac {1-x}{8}}={\tfrac {3}{4}}\cdot {\tfrac {1-x}{8}}\;\therefore \;x={\tfrac {3}{4}}\cdot {\tfrac {1-x}{8}}\;\therefore \;x={\tfrac {3}{35}};1-x={\tfrac {32}{35}}$ A son’s share = $\scriptstyle {\frac {7}{35}}$; the widow’s share= $\scriptstyle {\frac {4}{35}}$.
2. A son is entitled to receive twice as much as a daughter. Were there three sons and one daughter, each son would receive $\scriptstyle {\frac {2}{7}}$ths of the residue. Let $x$ be the stranger’s legacy.
   $\therefore {\tfrac {2}{7}}[1-x]=x\;\therefore \;x={\tfrac {2}{9}}\;{\text{and}}\;1-x={\tfrac {7}{9}}$
   Each Son’s share $={\tfrac {2}{5}}[1-x]={\tfrac {2}{5}}\times {\tfrac {7}{9}}={\tfrac {14}{45}}$
   The Daughter’s share $={\tfrac {1}{5}}[1-x]\ldots \ldots ={\tfrac {7}{45}}$
   The Stranger’s legacy $={\tfrac {2}{9}}\ldots \ldots \ldots \ldots ={\tfrac {10}{45}}$