Page:The Algebra of Mohammed Ben Musa (1831).djvu/189

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price of the measure is two and a half times as much as the sum which the donor has given the donee in money; and whenever the donee returns anything from the money capital, he returns from the measure as much as two and a half times that amount. Take now from the measure as much as corresponds to one thing, that is, two things and a half, and add this to what remains from the twenty, namely, twenty less thing. Thus the heirs of the deceased obtain twenty dirhems and one (112) thing and a half. The moiety of this is the legacy, namely, ten dirhems and three-fourths of thing; and this is one-third of the capital, namely, sixteen dirhems and two-thirds. Remove now ten dirhems on account of the opposite ten; there remain six dirhems and two-thirds, equal to three-fourths of thing. Complete the thing, by adding to it as much as one-third of the same; and add to the six dirhems and two-thirds

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It is arbitrary how he shall apportion this sum between the money capital and the measure.
If he pays on the money capital $p\cdot a$
and on the measure $q\cdot ma$
we have the equation $p\cdot a+q\cdot ma={\tfrac {2}{3}}ma$
or $p+q\;m={\tfrac {2}{3}}m$
The author assumes $p={\tfrac {m}{2}}\cdot q$
Whence $q={\tfrac {4}{9}}$, and $p={\tfrac {5}{9}}$, and therefore the donee pays
on the money capital ${\tfrac {5}{9}}a=11{\tfrac {1}{9}}$
and on the measure ${\tfrac {4}{9}}ma=22{\tfrac {2}{9}}$
Total $33{\tfrac {1}{3}}$