Pipes P and Q can fill a tank in 18 and 27 minutes, respectively, whereas pipe R can empty thefull tank in 54 minutes. P and Q were opened together for 6 minutes and then closed and R was opened. The tank was emptied by R alone in:

Options

A. 30 minutes
B. 40 minutes
C. 35 minutes
D. 45 minutes

Answer

To solve the problem step by step, we will first determine the rates at which pipes P, Q, and R work, and then calculate the total time taken by R to empty the tank after P and Q have filled it partially.

### Step 1: Determine the rates of pipes P, Q, and R

1. **Pipe P** can fill the tank in 18 minutes.
- Rate of P = 1 tank / 18 minutes = \( \frac{1}{18} \) tank per minute.

2. **Pipe Q** can fill the tank in 27 minutes.
- Rate of Q = 1 tank / 27 minutes = \( \frac{1}{27} \) tank per minute.

3. **Pipe R** can empty the tank in 54 minutes.
- Rate of R = 1 tank / 54 minutes = \( \frac{1}{54} \) tank per minute.

### Step 2: Find the combined rate of pipes P and Q

- Combined rate of P and Q:
\[ \text{Rate of P + Rate of Q} = \frac{1}{18} + \frac{1}{27} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 18 and 27 is 54.

- Convert the rates:
\[ \frac{1}{18} = \frac{3}{54}, \quad \frac{1}{27} = \frac{2}{54} \]

- Now add:
\[ \text{Combined rate} = \frac{3}{54} + \frac{2}{54} = \frac{5}{54} \text{ tanks per minute} \]

### Step 3: Calculate the amount of work done by P and Q in 6 minutes

- Work done in 6 minutes:
\[ \text{Work} = \text{Rate} \times \text{Time} = \frac{5}{54} \times 6 = \frac{30}{54} = \frac{5}{9} \text{ of the tank} \]

### Step 4: Determine how much work remains to be done by R

- Since P and Q filled \( \frac{5}{9} \) of the tank, the remaining work to empty is:
\[ 1 - \frac{5}{9} = \frac{4}{9} \text{ of the tank} \]

### Step 5: Calculate the time taken by R to empty the remaining work

- Rate of R is \( \frac{1}{54} \) tanks per minute.
- Time taken by R to empty \( \frac{4}{9} \) of the tank:
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{\frac{4}{9}}{\frac{1}{54}} = \frac{4}{9} \times 54 = 24 \text{ minutes} \]

### Final Answer

The tank was emptied by R alone in **24 minutes**.

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