A tap can fill a bath in 20 minutes and another tap can fill it in 30 minutes . Amit opens both the taps simultaneously. When the both should have been full, he finds that the waste pipe was open. He then closes the waste pipe and in another 4 minutes the bath is full. In what time would the waste pipe empty it?

Options

A. 35 minutes
B. 38 minutes
C. 36 minutes
D. 39 minutes

Answer

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

### Step 1: Determine the filling rates of the taps.
- Tap A fills the bath in 20 minutes, so its rate is:
\[ \text{Rate of A} = \frac{1 \text{ bath}}{20 \text{ minutes}} = \frac{3}{60} \text{ baths per minute} \]

- Tap B fills the bath in 30 minutes, so its rate is:
\[ \text{Rate of B} = \frac{1 \text{ bath}}{30 \text{ minutes}} = \frac{2}{60} \text{ baths per minute} \]

### Step 2: Calculate the combined filling rate of both taps.
- The combined rate when both taps are open is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} \text{ baths per minute} \]

### Step 3: Determine how long it would take to fill the bath without the waste pipe.
- To fill the entire bath (1 bath) at the combined rate:
\[ \text{Time to fill} = \frac{1 \text{ bath}}{\frac{5}{60} \text{ baths per minute}} = 12 \text{ minutes} \]

### Step 4: Understand the situation with the waste pipe.
- Amit finds out that the waste pipe was open. He closes it after some time, and in the next 4 minutes, the bath is full.
- This means that the bath was not filled completely in the first 12 minutes because of the waste pipe.

### Step 5: Calculate how much water was wasted during the 12 minutes.
- Let the rate of the waste pipe be \( C \) baths per minute. The effective filling rate with the waste pipe open is:
\[ \text{Effective Rate} = \text{Combined Rate} - C = \frac{5}{60} - C \]

- The time taken to fill the bath with the waste pipe open is 12 minutes, so:
\[ 12 \left( \frac{5}{60} - C \right) = \text{Amount filled in 12 minutes} \]

### Step 6: Calculate the amount filled in 12 minutes.
- In 12 minutes, the amount filled is:
\[ 12 \left( \frac{5}{60} - C \right) = 1 - \text{Amount wasted} \]

### Step 7: Calculate the amount filled in the last 4 minutes after closing the waste pipe.
- After closing the waste pipe, the filling rate is:
\[ \text{Rate without waste pipe} = \frac{5}{60} \]
- In 4 minutes, the amount filled is:
\[ 4 \times \frac{5}{60} = \frac{20}{60} = \frac{1}{3} \text{ baths} \]

### Step 8: Set up the equation for the total amount filled.
- The total amount filled in 12 minutes plus the amount filled in 4 minutes should equal 1 bath:
\[ 12 \left( \frac{5}{60} - C \right) + \frac{1}{3} = 1 \]

### Step 9: Solve for \( C \).
- Rearranging gives:
\[ 12 \left( \frac{5}{60} - C \right) = 1 - \frac{1}{3} = \frac{2}{3} \]
- Simplifying:
\[ 12 \left( \frac{1}{12} - C \right) = \frac{2}{3} \]
\[ 1 - 12C = \frac{2}{3} \]
\[ 12C = 1 - \frac{2}{3} = \frac{1}{3} \]
\[ C = \frac{1}{36} \text{ baths per minute} \]

### Step 10: Calculate the time taken by the waste pipe to empty the bath.
- The time taken to empty the bath is:
\[ \text{Time} = \frac{1 \text{ bath}}{C} = \frac{1}{\frac{1}{36}} = 36 \text{ minutes} \]

### Final Answer:
The waste pipe would empty the bath in **36 minutes**.