Page:The Algebra of Mohammed Ben Musa (1831).djvu/148

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you take the moiety from one-third, there remains one-sixth. This is the sum excepted. It is the share of the wife and the sister. Let it be five (thirteenth) parts. What remains of the one-third is five parts less one- sixth of the capital. The two-sevenths which he has excepted are two-sevenths of five parts less two-sevenths of one-sixth of the capital. Then you have six parts and three-sevenths, less one-sixth and two-sevenths of one-sixth of the capital. Add hereto two-thirds of the capital; then you have nineteen forty-seconds of the capital and six parts and three-sevenths, equal to thirteen parts. Subtract herefrom the six parts and three-sevenths. There remain nine-teen forty-seconds of the capital, equal to six parts and four-sevenths. Complete your capital by adding to it its double and four-nineteenths of it. Then you find the capital equal to fourteen parts, and seventy (98) one hundred and thirty-thirds of a part. Assume one part to be one hundred and thirty-three; then the whole capital is one thousand nine hundred and thirty-

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${\begin{aligned}&x+5v={\tfrac {1}{2}}\;;1-x+{\tfrac {2}{7}}[{\tfrac {1}{3}}-x]=13v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{3}}-x+{\tfrac {2}{7}}[{\tfrac {1}{3}}-x]=13v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {9}{7}}[{\tfrac {1}{3}}-x]=13v\\&\therefore \;{\tfrac {2}{3}}-{\tfrac {3}{14}}=[13-{\tfrac {45}{7}}]v\\&\therefore \;{\tfrac {19}{42}}={\tfrac {46}{7}}v\;\therefore \;{\tfrac {19}{276}}=v\\&\therefore \;x={\tfrac {29}{276}},\;{\text{and the residue}}\;={\tfrac {247}{276}}\\\end{aligned}}$
The author unnecessarily takes $7\times 276=1932$ for the common denominator.