Two inlet pipes, A and B, can fill an empty cistern in 22 and 33 hours, respectively. They were opened at the same time, but pipe A had to be closed 3 hours before the cistern was full. How many hours in total did it take the two pipes to fill the cistern?
(a)15
(b)17
(c)14.2
(d)16

Options

A. 15
B. 16.2
C. 14.2
D. 16

Answer

To solve the problem step by step, we need to determine how long it takes for both pipes A and B to fill the cistern together, considering that pipe A is closed 3 hours before the cistern is full.

### Step 1: Determine the rates of the pipes
- Pipe A can fill the cistern in 22 hours. Therefore, the rate of pipe A is:
\[ \text{Rate of A} = \frac{1}{22} \text{ cisterns per hour} \]
- Pipe B can fill the cistern in 33 hours. Therefore, the rate of pipe B is:
\[ \text{Rate of B} = \frac{1}{33} \text{ cisterns per hour} \]

### Step 2: Calculate the combined rate of both pipes
- The combined rate when both pipes are open is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{22} + \frac{1}{33} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 22 and 33 is 66.
\[ \frac{1}{22} = \frac{3}{66}, \quad \frac{1}{33} = \frac{2}{66} \]
- Therefore,
\[ \text{Combined Rate} = \frac{3}{66} + \frac{2}{66} = \frac{5}{66} \text{ cisterns per hour} \]

### Step 3: Determine the total time taken to fill the cistern
Let \( t \) be the total time taken to fill the cistern. Since pipe A is closed 3 hours before the cistern is full, it operates for \( t - 3 \) hours, while pipe B operates for the entire \( t \) hours.

- The amount of work done by pipe A in \( t - 3 \) hours is:
\[ \text{Work by A} = \left(t - 3\right) \times \frac{1}{22} \]
- The amount of work done by pipe B in \( t \) hours is:
\[ \text{Work by B} = t \times \frac{1}{33} \]

### Step 4: Set up the equation for total work
The total work done by both pipes together must equal 1 full cistern:
\[ \left(t - 3\right) \times \frac{1}{22} + t \times \frac{1}{33} = 1 \]

### Step 5: Solve the equation
Multiply through by 66 (the LCM of 22 and 33) to eliminate the fractions:
\[ 66 \left(t - 3\right) \times \frac{1}{22} + 66 t \times \frac{1}{33} = 66 \]
This simplifies to:
\[ 3(t - 3) + 2t = 66 \]
Expanding this gives:
\[ 3t - 9 + 2t = 66 \]
Combining like terms:
\[ 5t - 9 = 66 \]
Adding 9 to both sides:
\[ 5t = 75 \]
Dividing by 5:
\[ t = 15 \]

### Conclusion
The total time taken to fill the cistern is **15 hours**.