Page:The Algebra of Mohammed Ben Musa (1831).djvu/120

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On another Species of Legacies.

“A man dies, and leaves four sons, and bequeaths

to some person as much as the share of one of his sons; and to another, one-fourth of what remains after-the deduction of the above share from one-third.” You perceive that this legacy belongs to the class of those (77) which are taken from one-third of the capital.^[1] Computation: Take one-third of the capital, and subtract from it the share of a son. The remainder is one-third of the capital less the share. Then subtract from it one-fourth of what remains of the one-third, namely, one-fourth of one-third less one-fourth of the share. The remainder is one-fourth of the capital less three-fourths of the share. Add hereto two-thirds of the capital: then you have eleven-twelfths of the capital less three-fourths of a share, equal to four shares. Reduce this by removing the three-fourths of the share from the capital, and adding them to the four shares. Then you have eleven-twelfths of the capital, equal to four shares and three-fourths. Complete your capital, by adding to the four shares and three-fourths one-fourth of the same. Then you have five shares and two-elevenths,

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1. Let the first bequest $=v$; and the second $=y$
   Then $1-v-y=4v$
   i.e. ${\tfrac {2}{3}}+{\tfrac {3}{4}}-v-{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]=4v$
   $\therefore {\tfrac {2}{3}}+{\tfrac {3}{4}}[{\tfrac {1}{3}}-v]=4v$
   $\therefore {\tfrac {2}{3}}+{\tfrac {3}{12}}=[4+{\tfrac {3}{4}}]v\therefore {\tfrac {11}{12}}={\tfrac {19}{4}}v$
   $\therefore v={\tfrac {11}{57}}$; the 2d bequest $={\tfrac {2}{37}}$