Two pipes A and B can fill a cistern in 9 minutes and 12 minutes respectively. Both pipes are opened alternatively for 1 min each starting with A. In which time will they fill the cistern ?

A can fill a cistern in = 9 minutes $\Rightarrow$ 1 for 1 minute $=\frac{1}{9}$

B can fill a cistern in = 12 minutes $\Rightarrow$ for 1 minute $=\frac{1}{12}$

1. For 2 minutes both can fill $\Rightarrow \frac{1}{9}+\frac{1}{12}=\frac{4+3}{36}=\frac{7}{36}$ 

2. For another 2 minutes can fill $\Rightarrow 2\times \frac{7}{36}=\frac{14}{36}=\frac{7}{18}part$

3. For  another 2 minutes can fill  $\Rightarrow \frac{14}{36}+\frac{7}{36}=\frac{21}{36}$

4. For  another 2 minutes can fill $\Rightarrow \frac{28}{36}$

5. And another 2 minutes can fill  $\Rightarrow \frac{35}{36}$

$\text{Remaining}\Rightarrow 1-\frac{35}{36}=\frac{1}{36}$

$\frac{1}{36}partisfilledbyA\Rightarrow \frac{1}{36}\times 9\Rightarrow \frac{9}{36}=\frac{1}{4}\text{\hspace{0.17em}\hspace{0.17em}minutes}$

$\therefore Totaltime=5\times 2+\frac{1}{4}=10\frac{1}{4}\text{\hspace{0.17em}\hspace{0.17em}minutes}$