Two taps can fill a tub in 5 minutes and 7 minutes respectively. A pipe can empty it in 3 minutes. If all the three are kept open simultaneously, when will the tub be full ?

Question

Accepted Answer

To determine when the tub will be full when all three taps are open simultaneously, we need to calculate the net rate at which the tub is being filled.

1. **Rate of Filling by Tap 1:**
   - Tap 1 can fill the tub in 5 minutes.
   - Therefore, the rate of filling by Tap 1 is $ \frac{1}{5} $ of the tub per minute.

2. **Rate of Filling by Tap 2:**
   - Tap 2 can fill the tub in 7 minutes.
   - Therefore, the rate of filling by Tap 2 is $ \frac{1}{7} $ of the tub per minute.

3. **Rate of Emptying by the Pipe:**
   - The pipe can empty the tub in 3 minutes.
   - Therefore, the rate of emptying by the pipe is $ \frac{1}{3} $ of the tub per minute.

4. **Net Rate of Filling:**
   - The net rate of filling the tub when all three are open is the sum of the rates of the two taps minus the rate of the pipe:
   $$
   \text{Net rate} = \frac{1}{5} + \frac{1}{7} - \frac{1}{3}
   $$

5. **Calculating the Net Rate:**
   - To add these fractions, we find a common denominator, which is 105:
   $$
   \frac{1}{5} = \frac{21}{105}, \quad \frac{1}{7} = \frac{15}{105}, \quad \frac{1}{3} = \frac{35}{105}
   $$
   - Therefore, the net rate is:
   $$
   \frac{21}{105} + \frac{15}{105} - \frac{35}{105} = \frac{1}{105}
   $$

6. **Time to Fill the Tub:**
   - Since the net rate of filling the tub is $ \frac{1}{105} $ of the tub per minute, it will take 105 minutes to fill the tub completely.

Thus, the correct answer is Option 4: 105 minutes.

**Explanation of Incorrect Options:**

- **Option 1 (60 minutes):** This option is incorrect because the net rate of filling the tub is much slower than what would be required to fill the tub in 60 minutes. The calculations show that the tub fills at a rate of $ \frac{1}{105} $ per minute, not $ \frac{1}{60} $.

- **Option 2 (85 minutes):** This option is incorrect because, based on the net rate of filling, the tub cannot be filled in 85 minutes. The calculations clearly show that it takes 105 minutes.

- **Option 3 (90 minutes):** This option is incorrect because, similar to the other incorrect options, the net rate of filling does not support the tub being filled in 90 minutes. The correct time is 105 minutes.

By following the logical steps and calculations, we confirm that Option 4 is indeed the correct answer.