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:

$$\begin{gathered} q={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{9}}={\displaystyle \frac{17}{18}}\doteq 0.9444 \\ {q}_1=1-q=1-0.9444={\displaystyle \frac{1}{18}}\doteq 0.0556 \\ {n}_1=17 \\ {a}_1=n_1\mathrm{/}2=17\mathrm{/}2={\displaystyle \frac{17}{2}}=8.5 \\ {b}_1=n_1\mathrm{/}3=17\mathrm{/}3={\displaystyle \frac{17}{3}}\doteq 5.6667 \\ {c}_1=n_1\mathrm{/}9=17\mathrm{/}9={\displaystyle \frac{17}{9}}\doteq 1.8889 \\ {n}_2=1\mathrm{/}{q}_1=1\mathrm{/}0.0556=18 \\ a=n_2\mathrm{/}2=18\mathrm{/}2=9 \end{gathered}$$

$b=n_{2}/3=18/3=6$

$$\begin{gathered} c=n_2\mathrm{/}9=18\mathrm{/}9=2 \\ \text{ Verifying Solution: } \\ {n}_3=a+b+c=9+6+2=17 \end{gathered}$$

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