Four workers in a construction company

Question

There are four workers $A,B,C,D$ in a construction company with different skills. It takes $D_{AB}$ days when $A$ & $B$ works together to finish a construction, $D_{BC}$ days when $B$ & $C$ works together to finish the same construction and $D_{CD}$ days when $C$ & $D$ works together to finish the same construction. What is the maximum days it may take when all four workers work together to finish the same construction?

Answer 1

Let $\frac 1a, \frac 1b, \frac 1c, \frac 1d$ be the fraction of work $a,b,c,d$ do in a day, respectively. Then we have the following system of equations:

\begin{cases} \frac{D_{AB}}{a} + \frac{D_{AB}}{b} = 1 \\ \frac{D_{BC}}{b} + \frac{D_{BC}}{c} = 1 \\ \frac{D_{CD}}{c} + \frac{D_{CD}}{d} = 1 \end{cases}

By adding the first and third equation we get that $\frac 1a + \frac 1b + \frac 1c + \frac 1d = \frac{1}{D_{AB}} + \frac{1}{D_{CD}}$. So they will need $\frac{1}{\frac{1}{D_{AB}} + \frac{1}{D_{CD}}}$ days