Profit distribution workers

We have a profit of 40000e, and the work took 30 days. The first worker is promised 38% of the profit but only works for 15 days, the second worker is promised 33% of the profit and works for 20 days, and the third is promised 28% of the profit and works for 28 days. How do we fairly distribute the profit among the workers?

Final Answer:

a =  11165.524
b =  12928.5015
c =  15905.9745

Step-by-step explanation:

$$\begin{gathered} z=40000 \\ {p}_1=38\%=\frac{38}{100}=\frac{19}{50}=0.38 \\ {p}_2=33\%=\frac{33}{100}=0.33 \\ {p}_3=1-p_1-p_2=1-0.38-0.33=\frac{29}{100}=0.29 \\ {r}_1=\frac{15}{30}=\frac{1}{2}=0.5 \\ {r}_2=\frac{20}{30}=\frac{2}{3}\doteq 0.6667 \\ {r}_3=\frac{28}{30}=\frac{14}{15}\doteq 0.9333 \\ s=r_1+r_2+r_3=0.5+0.6667+0.9333=\frac{21}{10}=2\frac{1}{10}=2.1 \\ {k}_1=r_1\cdot p_1=0.5\cdot 0.38=\frac{19}{100}=0.19 \\ {k}_2=r_2\cdot p_2=0.6667\cdot 0.33=\frac{11}{50}=0.22 \\ {k}_3=r_3\cdot p_3=0.9333\cdot 0.29=\frac{203}{750}\doteq 0.2707 \\ K=1/(k_1+k_2+k_3)=1/(0.19+0.22+0.2707)\doteq 1.4691 \\ {P}_1=K\cdot k_1=1.4691\cdot 0.19=\frac{285}{1021}\doteq 0.2791 \\ {P}_2=K\cdot k_2=1.4691\cdot 0.22=\frac{330}{1021}\doteq 0.3232 \\ {P}_3=K\cdot k_3=1.4691\cdot 0.2707=\frac{406}{1021}\doteq 0.3976 \\ a=P_1\cdot z=0.2791\cdot 40000=11165.524 \end{gathered}$$

$b=P_2\cdot z=0.3232\cdot 40000=12928.5015$

$$\begin{gathered} c=P_3\cdot z=0.3976\cdot 40000\doteq 15905.9745 \\ \text{ Verifying Solution: } \\ S=a+b+c=11165.524+12928.5015+15905.9745=40000\text{eur} \end{gathered}$$

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