# A farmer's heritage

A farmer died leaving his 17 horses to his 3 sons.
When his sons opened up the Will it read:
My eldest son should get 1/2 (half) of total horses;
My middle son should be given 1/3rd (one-third) of the total horses;
My youngest son should be given 1/9th (one-ninth) of the total horses.

How many horses should each son-heir get?

Correct result:

a =  9
b =  6
c =  2

#### Solution:

$q=\dfrac{ 1 }{ 2 } +\dfrac{ 1 }{ 3 } +\dfrac{ 1 }{ 9 }=\dfrac{ 17 }{ 18 } \doteq 0.9444 \ \\ q_{1}=1 - q=1 - 0.9444 \doteq \dfrac{ 1 }{ 18 } \doteq 0.0556 \ \\ n_{1}=17 \ \\ a_{1}=n_{1}/2=17/2=\dfrac{ 17 }{ 2 }=8.5 \ \\ b_{1}=n_{1}/3=17/3 \doteq \dfrac{ 17 }{ 3 } \doteq 5.6667 \ \\ c_{1}=n_{1}/9=17/9 \doteq \dfrac{ 17 }{ 9 } \doteq 1.8889 \ \\ \ \\ n_{2}=1/q_{1}=1/0.0556=18 \ \\ a=n_{2}/2=18/2=9$
$b=n_{2}/3=18/3=6$
$c=n_{2}/9=18/9=2 \ \\ \ \\ \text{ Correctness test: } \ \\ n_{3}=a+b+c=9+6+2=17$

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