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 ?

Options

A. 60 min
B. 85 min
C. 90 min
D. 105 min

Answer

To solve the problem of how long it will take to fill the tub when two taps and one pipe are working simultaneously, we can follow these steps:

### Step 1: Determine the rates of filling and emptying
- Tap A fills the tub in 5 minutes, so its rate is \( \frac{1}{5} \) of the tub per minute.
- Tap B fills the tub in 7 minutes, so its rate is \( \frac{1}{7} \) of the tub per minute.
- The pipe empties the tub in 3 minutes, so its rate is \( \frac{1}{3} \) of the tub per minute (but since it's emptying, we will consider this rate as negative).

### Step 2: Calculate the combined rate of filling
The combined rate of filling when both taps are open and the pipe is emptying is:
\[ \text{Combined Rate} = \left(\frac{1}{5} + \frac{1}{7} - \frac{1}{3}\right) \]

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 5, 7, and 3 is 105.

### Step 4: Convert each rate to the common denominator
- For \( \frac{1}{5} \):
\[ \frac{1}{5} = \frac{21}{105} \]
- For \( \frac{1}{7} \):
\[ \frac{1}{7} = \frac{15}{105} \]
- For \( \frac{1}{3} \):
\[ \frac{1}{3} = \frac{35}{105} \]

### Step 5: Substitute back into the combined rate
Now substituting these values into the combined rate:
\[ \text{Combined Rate} = \frac{21}{105} + \frac{15}{105} - \frac{35}{105} \]
\[ = \frac{21 + 15 - 35}{105} = \frac{1}{105} \]

### Step 6: Calculate the time to fill the tub
Since the combined rate of filling is \( \frac{1}{105} \) of the tub per minute, it will take:
\[ \text{Time} = \frac{1 \text{ tub}}{\frac{1}{105} \text{ tub/min}} = 105 \text{ minutes} \]

### Final Answer
The tub will be full in **105 minutes**.

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