Al-Khwarizmi/The Algebra/Inheritance Problem
Problem
- A woman dies, leaving her husband, a son and three daughters.
- She also leaves $\frac 1 8 + \frac 1 7$ of her estate to a stranger.
- According to law:
- the husband receives one quarter of the estate
- the son receives double the share of a daughter
- but this division is made only after the legacy to the stranger has been paid.
How should the inheritance be divided?
Solution
The stranger gets $300$ of $1120$ parts of the legacy.
The husband gets $205$ of $1120$ parts of the legacy.
The son gets $246$ of $1120$ parts of the legacy.
The daughters each get $123$ of $1120$ parts of the legacy.
Proof
Let $x$ be the total legacy.
The stranger gets $x \paren {\dfrac 1 7 + \dfrac 1 8} = \dfrac {15 x} {56}$.
This leaves $x \paren {1 - \dfrac {15} {56} } = \dfrac {41 x} {56}$ to be divided between the family.
The husband gets $\dfrac 1 4 \times \dfrac {41 x} {56}$
The son gets twice what the daughters get, so the remaining $\dfrac 3 4 \times \dfrac {41 x} {56}$ gets split $5$ ways:
- $\dfrac 1 5 \times \dfrac 3 4 \times \dfrac {41 x} {56}$ to each of the $3$ daughters
- $\dfrac 2 5 \times \dfrac 3 4 \times \dfrac {41 x} {56}$ to the son.
To expressing each share in the same denominator, we make sure each fraction is a multiple of $\dfrac 1 {20 \times 56} = \dfrac 1 {1120}$.
Hence:
- the stranger's portion: $\dfrac {20 \times 15 x} {1120} = \dfrac {300 x} {1120}$
- the husband's portion: $\dfrac {5 \times 41 x} {1120} = \dfrac {205 x} {1120}$
- the son's portion: $\dfrac {2 \times 3 \times 41 x} {1120} = \dfrac {246 x} {1120}$
- each daughter's portion: $\dfrac {1 \times 3 \times 41 x} {1120} = \dfrac {123 x} {1120}$
$\blacksquare$
Sources
- c. 830: Muhammad ibn Musa al-Khwarizmi: Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Al-Khwarizmi (c. $825$ ad): $37$