Master 7

A master craftsman and an apprentice are laying tiles on a roof. At the end of the work, they found that the apprentice did only one third of the work and the master did the rest. If the master worked alone, the work would take him 2 hours longer than if they worked together. If the apprentice worked alone, the work would take him 8 hours longer than if they worked together. How long would it take the master alone and the apprentice alone to complete the work?

Final Answer:

a =  6 h
b =  12 h

Step-by-step explanation:

$$\begin{gathered} s=1/(1/a+1/b) \\ (s+2)\cdot (1/a)=1 \\ (s+8)\cdot (1/b)=1 \\ A=1/b;B=1/b \\ s\cdot A+s\cdot B=1 \\ (s+2)\cdot A=1 \\ (s+8)\cdot B=1 \\ s\cdot A+s\cdot B=1 \\ s\cdot A+2\cdot A=1 \\ s\cdot B+8\cdot B=1 \\ sA+sB-sA-2A-sB-8B=1-1-1 \\ -2A-8B=-1 \\ 2A+8B=1 \\ s=4\text{h} \\ A=1/(s+2)=1/(4+2)=\frac{1}{6}\doteq 0.1667 \\ B=1/(s+8)=1/(4+8)=\frac{1}{12}\doteq 0.0833 \\ a=1/A=1/0.1667=6\text{h} \end{gathered}$$

$b=1/B=1/0.0833=12\text{h}$

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