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Two persons having different productivity of labour working together can reap a field in 2 days. If one-third of the field was reaped by the first man and the rest by the other one working alternatively took 4 days. How long did it take for the faster person to reap the whole field working alone?

Can anyone please explain me this solution

total Efficiency of two people is $50%$. Ratio of efficiency is $1:2$ Efficiency of second person is $33.33%$

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Let $x,y$ the respective times (in days) in harvesting the field separately.

The respective productivities:$\dfrac{1}{x}, \dfrac{1}{y}$. Then: $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{2}\rightarrow 2x+2y-xy=0$

$\dfrac{1}{x}=\dfrac{\frac{1}{3}}{t}\rightarrow x=3t$ and $\dfrac{1}{y}=\dfrac{\frac{2}{3}}{4-t}\rightarrow y=\dfrac{3}{2}(4-t)$. Substituting:

$$6t+3(4-t)-\dfrac{3}{2}(4-t)3t=0\rightarrow 12t+24-6t-36t+9t^{2}=0\rightarrow$$

$$9t^{2}-30t+24=0\rightarrow 3t^{2}-10t+8=0\rightarrow \left\{ \begin{array}{lcc} t=2\rightarrow \left\{ \begin{array}{lcc} x=6 \\ \\ y=3 \end{array} \right.\\ \\ t=\frac{4}{3}\rightarrow \left\{ \begin{array}{lcc} x=4 \\ \\ y=4 \end{array} \right. \end{array} \right.$$ The answer is: $6$ days

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This answer is clearly not right. It can't take the faster person longer that it takes a combination of the slower and faster person. The answer is 3 days.

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