Three pipes A, B and C can fill a tank from empty to full in 30 minutes, 20 minutes and 10 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 R in the liquid in the tank after 3 minutes?
A)`(3)/(11)`
B)`(6)/(11)`
C)`(4)/(11)`
D)`(7)/(11)`

Options

A. `(3)/(11)`
B. `(6)/(11)`
C. `(4)/(11)`
D. `(7)/(11)`

Answer

To solve the problem, we need to determine the amount of each solution (P, Q, and R) in the tank after 3 minutes when all three pipes A, B, and C are opened.

### Step 1: Determine the filling rates of each pipe
- Pipe A can fill the tank in 30 minutes. Therefore, in 1 minute, it fills \( \frac{1}{30} \) of the tank.
- Pipe B can fill the tank in 20 minutes. Therefore, in 1 minute, it fills \( \frac{1}{20} \) of the tank.
- Pipe C can fill the tank in 10 minutes. Therefore, in 1 minute, it fills \( \frac{1}{10} \) of the tank.

### Step 2: Calculate the total filling rate of all three pipes
To find the total rate at which the tank is filled when all three pipes are opened, we add their individual rates:
\[ \text{Total rate} = \frac{1}{30} + \frac{1}{20} + \frac{1}{10} \]

To add these fractions, we need a common denominator. The least common multiple of 30, 20, and 10 is 60. Therefore, we convert each fraction:
\[ \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{10} = \frac{6}{60} \]
Now, adding these gives:
\[ \text{Total rate} = \frac{2}{60} + \frac{3}{60} + \frac{6}{60} = \frac{11}{60} \]

### Step 3: Calculate the amount of tank filled in 3 minutes
In 3 minutes, the amount of the tank filled is:
\[ \text{Amount filled} = \text{Total rate} \times \text{Time} = \frac{11}{60} \times 3 = \frac{33}{60} = \frac{11}{20} \]

### Step 4: Determine the amount of each solution in the tank
Now, we need to find out how much of each solution (P, Q, R) is in the tank after 3 minutes:
- The amount of solution P from pipe A (which fills at \( \frac{1}{30} \) per minute):
\[ \text{Amount of P} = \frac{1}{30} \times 3 = \frac{3}{30} = \frac{1}{10} \]
- The amount of solution Q from pipe B (which fills at \( \frac{1}{20} \) per minute):
\[ \text{Amount of Q} = \frac{1}{20} \times 3 = \frac{3}{20} \]
- The amount of solution R from pipe C (which fills at \( \frac{1}{10} \) per minute):
\[ \text{Amount of R} = \frac{1}{10} \times 3 = \frac{3}{10} \]

### Step 5: Calculate the total amount of solutions in the tank
Now we sum the amounts of P, Q, and R:
\[ \text{Total amount} = \frac{1}{10} + \frac{3}{20} + \frac{3}{10} \]
To add these, we convert them to a common denominator, which is 20:
\[ \frac{1}{10} = \frac{2}{20}, \quad \frac{3}{10} = \frac{6}{20} \]
Thus,
\[ \text{Total amount} = \frac{2}{20} + \frac{3}{20} + \frac{6}{20} = \frac{11}{20} \]

### Step 6: Find the proportion of solution R
Now, we find the proportion of solution R in the total liquid:
\[ \text{Proportion of R} = \frac{\text{Amount of R}}{\text{Total amount}} = \frac{\frac{3}{10}}{\frac{11}{20}} = \frac{3}{10} \times \frac{20}{11} = \frac{6}{11} \]

### Final Answer
The proportion of solution R in the liquid in the tank after 3 minutes is \( \frac{6}{11} \).