Three pipes, A, B and C, can fill a tank from empty to full in 40 minutes, 20 minutes and 30 minutes, respectively. When the tank is empty, all the three pipes are opened. A, B and C discharge chemical solutions P, Q and R, respectively. What is the proportion of solution Q in the liquid in the tank after 9 minutes?

The correct option is 6/13.

Step 1: Understand the rate of work for each pipe
Let the total capacity of the tank be 1 unit (or any unit of volume).

Pipe A can fill the tank in 40 minutes.
Therefore, the fraction of the tank filled by Pipe A in 1 minute =

$\frac{1}{40}$

Pipe B can fill the tank in 20 minutes.
Therefore, the fraction of the tank filled by Pipe B in 1 minute =

$\frac{1}{20}$

Pipe C can fill the tank in 30 minutes.
Therefore, the fraction of the tank filled by Pipe C in 1 minute =

$\frac{1}{30}$

Step 2: Calculate the total liquid entering the tank per minute
When all three pipes are opened together, the total fraction of the tank filled by A, B, and C combined in 1 minute is:

$\text{Total rate}=\frac{1}{40}+\frac{1}{20}+\frac{1}{30}$

To add these fractions, find the Least Common Multiple (LCM) of the denominators (40, 20, 30), which is 120:

$\text{Total rate}=\frac{3+6+4}{120}=\frac{13}{120}$

So, in 1 minute, $\frac{13}{120}$ of the tank is filled with liquid.

Step 3: Determine the proportion of chemical solution Q
Chemical solution Q is discharged by Pipe B. The proportion of solution Q in the total liquid in the tank at any given time (including after 9 minutes) is simply the ratio of Pipe B's filling rate to the combined filling rate of all three pipes.

$\text{Proportion of Q}=\frac{\text{Rate of Pipe B}}{\text{Total combined rate}}$

$\text{Proportion of Q}=\frac{1/20}{13/120}$

$\text{Proportion of Q}=\frac{1}{20}\times \frac{120}{13}=\frac{6}{13}$

Alternatively, in 9 minutes:
- Amount of solution Q discharged by Pipe B = $9\times \frac{1}{20}=\frac{9}{20}$
- Total liquid in the tank = $9\times \frac{13}{120}=\frac{117}{120}$

The proportion of solution Q is:

$\frac{9/20}{117/120}=\frac{9}{20}\times \frac{120}{117}=\frac{9\times 6}{117}=\frac{54}{117}=\frac{6}{13}$

Thus, the proportion of solution Q in the liquid in the tank after 9 minutes is 6/13.