Three taps A, B and C can fill a tank in 120, 80 and 96 minutes respectively. If all the taps are opened together, then in how many minutes will the tank be filled?

Options

A. 35
B. 32
C. 43
D. 42

Answer

To solve the problem of how long it will take to fill the tank when all three taps A, B, and C are opened together, we can follow these steps:

### Step 1: Determine the filling rates of each tap
- Tap A fills the tank in 120 minutes. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{120} \text{ tank/minute} \]

- Tap B fills the tank in 80 minutes. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{80} \text{ tank/minute} \]

- Tap C fills the tank in 96 minutes. Therefore, its rate is:
\[ \text{Rate of C} = \frac{1}{96} \text{ tank/minute} \]

### Step 2: Calculate the combined filling rate
To find the combined filling rate when all taps are opened together, we add their individual rates:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the rates:
\[ \text{Combined Rate} = \frac{1}{120} + \frac{1}{80} + \frac{1}{96} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 120, 80, and 96 is 480. We will convert each rate to have this common denominator:
\[ \frac{1}{120} = \frac{4}{480}, \quad \frac{1}{80} = \frac{6}{480}, \quad \frac{1}{96} = \frac{5}{480} \]

### Step 4: Add the rates
Now, we can add the rates:
\[ \text{Combined Rate} = \frac{4}{480} + \frac{6}{480} + \frac{5}{480} = \frac{15}{480} \]

### Step 5: Simplify the combined rate
We can simplify the combined rate:
\[ \text{Combined Rate} = \frac{15}{480} = \frac{1}{32} \text{ tank/minute} \]

### Step 6: Calculate the time to fill the tank
To find the time taken to fill one tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{32} \text{ tank/minute}} = 32 \text{ minutes} \]

### Final Answer
Thus, if all the taps are opened together, the tank will be filled in **32 minutes**.

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