Bull grazing periods

I need to find out the number of bulls in the third period. In the first period, 12 bulls grazed on 3 and 1/3 ha for 4 weeks. In the second period, 21 bulls graze on 10 ha for 9 weeks. How many bulls in the third period graze for 24ha and 18 weeks? The pasture grows evenly.

Final Answer:

c =  36

Step-by-step explanation:

$$\begin{gathered} a=12 \\ {S}_1=3\frac{1}{3}=3+\frac{1}{3}=\frac{3\cdot 3+1}{3}=\frac{9+1}{3}=\frac{10}{3}=3\frac{1}{3}\doteq 3.3333\text{ha} \\ {t}_1=4 \\ b=21 \\ {S}_2=10\text{ha} \\ {t}_2=9 \\ {S}_3=24\text{ha} \\ {t}_3=18 \\ {S}_1-a\cdot X\cdot t_1+S_1\cdot t_1\cdot Y=0 \\ {S}_2-b\cdot X\cdot t_2+S_2\cdot t_2\cdot Y=0 \\ 3.3333333333333-12\cdot X\cdot 4+3.3333333333333\cdot 4\cdot Y=0 \\ 10-21\cdot X\cdot 9+10\cdot 9\cdot Y=0 \\ 48X-13.333333Y=3.333333 \\ 189X-90Y=10 \\ Pivot:Row1\leftrightarrow Row2 \\ 48X-13.33Y=3.33 \\ Row2-\frac{48}{189}\cdot Row1\to Row2 \\ 9.52Y=0.79 \\ Y=\frac{0.79365079}{9.52380952}=0.08333333 \\ X=\frac{10+90Y}{189}=\frac{10+90\cdot 0.08333333}{189}=0.09259259 \\ X=\frac{5}{54}\doteq 0.092593 \\ Y=\frac{1}{12}\doteq 0.083333 \\ {S}_3-c\cdot X\cdot t_3+S_3\cdot t_3\cdot Y=0 \\ 24-c\cdot 0.09259259259259\cdot 18+24\cdot 18\cdot 0.083333333333329=0 \\ 1.666667c=60 \\ c=\frac{60}{1.66666667}=36 \\ c=36 \end{gathered}$$

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