Page:The Algebra of Mohammed Ben Musa (1831).djvu/121

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equal to the capital. Suppose, now, every share to be eleven; then the whole square will be fifty-seven; one-third of this is nineteen; from this one share, namely, eleven, must be subtracted; there remain eight. The legatee, to whom one-fourth of this remainder was bequeathed, receives two. The remaining six are returned to the other two-thirds, which are thirty-eight. Their sum is forty-four, which is to be divided amongst the four sons; so that each son receives eleven.

If he leaves four sons, and bequeaths to a person as much as the share of a son, less one-fifth of what remains from one-third after the deduction of that share, then this is likewise a legacy, which is taken from one-third.^[1] Take one-third, and subtract from it one share; there remains one-third less the share. Then return to it that which was excepted, namely, one-fifth of the one-third less one-fifth of the share. This gives one-third and one-fifth of one-third (or two-fifths) (78) less one share and one-fifth of a share. Add this to two-thirds of the capital. The sum is, the capital and one-third of one-fifth of the capital less one share and one-fifth of a share, equal to four-shares. Reduce this by removing one share and one-fifth from the capital,

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1. $1-v+{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]=4v$
   or ${\tfrac {2}{3}}\times {\tfrac {1}{3}}-v+{\tfrac {1}{5}}[{\tfrac {1}{3}}-v]=4v$
   or ${\tfrac {2}{3}}+{\tfrac {6}{5}}[{\tfrac {1}{3}}-v]=4v$
   $\therefore \;{\tfrac {2}{3}}+{\tfrac {2}{5}}=[4+{\tfrac {6}{5}}]v\;\therefore \;{\tfrac {16}{15}}={\tfrac {26}{5}}v$
   $\therefore \;v={\tfrac {8}{39}}$, and the stranger’s lcgacy $={\tfrac {7}{39}}$

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