A farmer's heritage

A farmer died, leaving his 17 horses to his three sons.
When his sons opened up the testament and read:
My eldest son should get 1/2 (half) of the 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?

Final Answer:

a =  9
b =  6
c =  2

Step-by-step explanation:

$$\begin{gathered} q=\frac{1}{2}+\frac{1}{3}+\frac{1}{9}=\frac{9}{18}+\frac{6}{18}+\frac{2}{18}=\frac{9+6+2}{18}=\frac{17}{18}\doteq 0.9444 \\ {q}_1=1-q=1-\frac{17}{18}=\frac{1\cdot 18}{18}-\frac{17}{18}=\frac{18}{18}-\frac{17}{18}=\frac{18-17}{18}=\frac{1}{18}\doteq 0.0556 \\ {n}_1=17 \\ {a}_1=n_1/2=17/2=\frac{17}{2}=8\frac{1}{2}=8.5 \\ {b}_1=n_1/3=17/3=\frac{17}{3}=5\frac{2}{3}\doteq 5.6667 \\ {c}_1=n_1/9=17/9=\frac{17}{9}=1\frac{8}{9}\doteq 1.8889 \\ {n}_2=1/q_1=1/\frac{1}{18}=1:\frac{1}{18}=1\cdot \frac{18}{1}=\frac{1\cdot 18}{1}=\frac{18}{1}=18 \\ a=n_2/2=18/2=9 \end{gathered}$$

$b=n_2/3=18/3=6$

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

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