Work men and women

Two men and 1 woman can do a piece of work in 14 days, while 4 women and 2 men can do the same work in 8 days. If a man get 90 euros/day, what should be the wages/day of a woman?

Correct answer:

w =  60 Eur

Step-by-step explanation:

$$\begin{gathered} m=90\text{eur/d} \\ 14(2/a+1/b)=1 \\ 8(4/b+2/a)=1 \\ 14(2A+1B)=1 \\ 8(4B+2A)=1 \\ 14\cdot (2\cdot A+1\cdot B)=1 \\ 8\cdot (4\cdot B+2\cdot A)=1 \\ 28A+14B=1 \\ 16A+32B=1 \\ Row2-\frac{16}{28}\cdot Row1\to Row2 \\ 28A+14B=1 \\ 24B=0.43 \\ B=\frac{0.42857143}{24}=0.01785714 \\ A=\frac{1-14B}{28}=\frac{1-14\cdot 0.01785714}{28}=0.02678571 \\ A=\frac{3}{112}\doteq 0.026786 \\ B=\frac{1}{56}\doteq 0.017857 \\ a=1/A=1/0.0268=\frac{112}{3}\doteq 37.3333\text{d} \\ b=1/B=1/0.0179=56\text{d} \\ w=\frac{a}{b}\cdot m=\frac{37.3333}{56}\cdot 90=60\text{Eur} \end{gathered}$$

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