Two taps can separately fill a cistern in 10 minutes and 15 minutes , respectively and when the waste pipe is open , they can together fill it in 18 mintutes . 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 need to find out how long the waste pipe takes to empty the full cistern.

### Step 1: Determine the filling rates of the taps
- Tap A fills the cistern in 10 minutes.
- Tap B fills the cistern in 15 minutes.

To find their rates:
- Rate of Tap A = 1 cistern / 10 minutes = 1/10 cistern per minute
- Rate of Tap B = 1 cistern / 15 minutes = 1/15 cistern per minute

### Step 2: Find the combined filling rate of both taps
- Combined rate of Tap A and Tap B = (1/10 + 1/15) cistern per minute.

To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30.
- (1/10) = 3/30
- (1/15) = 2/30

So,
- Combined rate = (3/30 + 2/30) = 5/30 = 1/6 cistern per minute.

### Step 3: Determine the combined rate with the waste pipe open
When the waste pipe is open, the taps can fill the cistern in 18 minutes.
- Rate with waste pipe = 1 cistern / 18 minutes = 1/18 cistern per minute.

### Step 4: Set up the equation
Let the rate of the waste pipe be C (in cisterns per minute). The equation is:
- Combined rate of taps - Rate of waste pipe = Rate with waste pipe open
- (1/10 + 1/15) - C = 1/18

Substituting the combined rate:
- (1/6) - C = 1/18

### Step 5: Solve for C
Rearranging the equation gives:
- C = (1/6) - (1/18)

To subtract these fractions, we need a common denominator. The LCM of 6 and 18 is 18.
- (1/6) = 3/18

So,
- C = (3/18) - (1/18) = 2/18 = 1/9 cistern per minute.

### Step 6: Find the time taken by the waste pipe to empty the cistern
If the waste pipe empties at a rate of 1/9 cistern per minute, then the time taken to empty 1 full cistern is:
- Time = 1 cistern / (1/9 cistern per minute) = 9 minutes.

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

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