# 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:
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