Taps X and Y can fill a tank in 30 and 40 minutes respectively.Tap Z can empty the filled tank in 60 minutes.If all the three taps are kept open for one minute each,how much time will the taps take to fill the tank?

Options

A. A) 48min
B. B) 72min
C. C) 24min
D. D) None of these

Answer

To solve the problem step by step, we will calculate the rates at which each tap fills or empties the tank and then combine these rates to find out how long it will take to fill the tank when all taps are open.

### Step 1: Determine the filling rate of Tap X
Tap X can fill the tank in 30 minutes. Therefore, the part of the tank filled by Tap X in 1 minute is:
\[ \text{Rate of Tap X} = \frac{1}{30} \text{ tank per minute} \]

### Step 2: Determine the filling rate of Tap Y
Tap Y can fill the tank in 40 minutes. Therefore, the part of the tank filled by Tap Y in 1 minute is:
\[ \text{Rate of Tap Y} = \frac{1}{40} \text{ tank per minute} \]

### Step 3: Determine the emptying rate of Tap Z
Tap Z can empty the filled tank in 60 minutes. Therefore, the part of the tank emptied by Tap Z in 1 minute is:
\[ \text{Rate of Tap Z} = -\frac{1}{60} \text{ tank per minute} \]
(Note: The rate is negative because it is emptying the tank.)

### Step 4: Combine the rates of all taps
Now, we can combine the rates of all three taps to find the net rate of filling the tank when all taps are open:
\[ \text{Net Rate} = \text{Rate of Tap X} + \text{Rate of Tap Y} + \text{Rate of Tap Z} \]
Substituting the values:
\[ \text{Net Rate} = \frac{1}{30} + \frac{1}{40} - \frac{1}{60} \]

### Step 5: Find a common denominator
The least common multiple (LCM) of 30, 40, and 60 is 120. We will convert each fraction to have this common denominator:
\[ \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120}, \quad \frac{1}{60} = \frac{2}{120} \]
Now substituting these values back into the equation:
\[ \text{Net Rate} = \frac{4}{120} + \frac{3}{120} - \frac{2}{120} = \frac{5}{120} \]

### Step 6: Simplify the net rate
\[ \text{Net Rate} = \frac{5}{120} = \frac{1}{24} \text{ tank per minute} \]

### Step 7: Calculate the time to fill the tank
If the net rate is \(\frac{1}{24}\) of the tank per minute, then the time taken to fill the entire tank is the reciprocal of the net rate:
\[ \text{Time to fill the tank} = \frac{1}{\frac{1}{24}} = 24 \text{ minutes} \]

### Final Answer
The taps will take **24 minutes** to fill the tank when all three taps are kept open.

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