Two taps can separately fill a cistern in 10 minutes and 15 minutes respectively and when the waste pipe is opened , they can together fill it in 18 minutes . The waste pipe can empty the full cistern in

Options

A. 7 min
B. 9 min
C. 13 min
D. 23 min

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the total work done
We need to find the least common multiple (LCM) of the times taken by the two taps and the time taken when the waste pipe is opened. The taps can fill the cistern in 10 minutes and 15 minutes, and together they can fill it in 18 minutes.

- LCM of 10, 15, and 18:
- Prime factorization:
- 10 = 2 × 5
- 15 = 3 × 5
- 18 = 2 × 3²
- LCM = 2 × 3² × 5 = 90

So, the total work (TW) is 90 units.

### Step 2: Calculate the efficiency of each tap and the waste pipe
- Efficiency of Tap A (fills in 10 minutes):
\[ \text{Efficiency of A} = \frac{90 \text{ units}}{10 \text{ minutes}} = 9 \text{ units/minute} \]

- Efficiency of Tap B (fills in 15 minutes):
\[ \text{Efficiency of B} = \frac{90 \text{ units}}{15 \text{ minutes}} = 6 \text{ units/minute} \]

- Efficiency of both taps together (fills in 18 minutes):
\[ \text{Efficiency of A + B} = \frac{90 \text{ units}}{18 \text{ minutes}} = 5 \text{ units/minute} \]

### Step 3: Calculate the efficiency of the waste pipe
When both taps are open, they fill the cistern at a combined rate of:
\[ \text{Efficiency of A + B} = 9 + 6 = 15 \text{ units/minute} \]

Since they can fill the cistern together with the waste pipe at 5 units/minute, we can find the efficiency of the waste pipe (C):
\[ \text{Efficiency of C} = \text{Efficiency of A + B} - \text{Efficiency of A + B with waste pipe} \]
\[ \text{Efficiency of C} = 15 - 5 = 10 \text{ units/minute} \]

### Step 4: Calculate the time taken by the waste pipe to empty the cistern
To find the time taken by the waste pipe to empty the full cistern, we can use the efficiency we just calculated:
\[ \text{Time taken by C} = \frac{\text{Total Work}}{\text{Efficiency of C}} = \frac{90 \text{ units}}{10 \text{ units/minute}} = 9 \text{ minutes} \]

### Final Answer
The waste pipe can empty the full cistern in **9 minutes**.

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