Time and Work problem 2 by Dhaval Furia

Let the work to be done be $W$ , and the rates at which $A$ and $B$ work be $r_A$ and $r_B$ work/day respectively. Then we have

$\begin{cases} 12(r_A + r_B) = W & ...(1) \\ 9\left(\dfrac {r_A}2 + 3r_B\right) = W & ...(2) \end{cases}$

$\begin{aligned} 9 \times (1) - 4 \times (2): \quad (108-18)r_A & = 5W \\ 90r_A & = 5W \\ \implies \boxed{18} r_A & = W \end{aligned}$

Therefore $A$ takes $\boxed{18}$ days to complete the work $W$ .

Let with usual efficiency, $A$ does the work in $x$ days and $B$ does it in $y$ days. Then

$\dfrac{xy}{x+y}=12$

$\dfrac{2x\times \frac{y}{3}}{2x+\frac{y}{3}}=\dfrac{2xy}{6x+y}=9$

So, $y=2x$

Substituting in the first equation, we get $x=\boxed {18}$ .