Collecting nuts

Ondra, Mathias, and Kuba are returning from collecting nuts. They have a total of 120. Mathias complains that Ondra has the most, as always. The father orders Ondra to sprinkle it on his Mathias so that the number of nuts doubles. Now Cuba is complaining that he has the most.
At his father's command, Mathias sprinkles Kuba with double the number of nuts. Ondra is angry that he has the fewest now, so Cuba sprinkles Ondra with double the number of nuts. Now everyone has the same, and finally, they are calm.

How many nuts did each of the boys initially have?

Final Answer:

o =  55
m =  35
k =  30

Step-by-step explanation:

$$\begin{gathered} o+m+k=120 \\ {o}_1=o-m \\ {m}_1=m+m=2m \\ {k}_1=k \\ {m}_2=m_1-k_1=2m-k \\ {k}_2=k_1+k_1=2k_1=2k \\ {o}_2=o_1 \\ {o}_3=o_2+o_2=o_1+o_1=(o-m)+(o-m) \\ {k}_3=k_2-o_2=2k-o_1=2k-(o-m) \\ {m}_3=m_2 \\ o+m+k=120 \\ 2k-(o-m)=120/3 \\ (o-m)+(o-m)=120/3 \\ 2\cdot k-(o-m)=120/3 \\ (o-m)+(o-m)=120/3 \\ k+m+o=120 \\ 6k+3m-3o=120 \\ 6m-6o=-120 \\ Pivot:Row1\leftrightarrow Row2 \\ Row2-\frac{1}{6}\cdot Row1\to Row2 \\ 0.5m+1.5o=100 \\ Pivot:Row2\leftrightarrow Row3 \\ Row3-\frac{0.5}{6}\cdot Row2\to Row3 \\ 2o=110 \\ o=\frac{110}{2}=55 \\ m=\frac{-120+6o}{6}=\frac{-120+6\cdot 55}{6}=35 \\ k=\frac{120-3m+3o}{6}=\frac{120-3\cdot 35+3\cdot 55}{6}=30 \\ k=30 \\ m=35 \\ o=55 \end{gathered}$$

$m=35$

$$\begin{gathered} k=30 \\ \text{ Verifying Solution: } \\ {o}_1=o-m=55-35=20 \\ {m}_1=m+m=35+35=70 \\ {k}_1=k=30 \\ {o}_2=o_1=20 \\ {m}_2=m_1-k_1=70-30=40 \\ {k}_2=k_1+k_1=30+30=60 \\ {o}_3=o_2+o_2=20+20=40 \\ {m}_3=m_2=40 \\ {k}_3=k_2-o_2=60-20=40 \end{gathered}$$

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