Three taps A, B and C can fill an tank in 50, 60 and 75 hours respectively. How long (in hours) would the three taps take to fill the tank if all of them are opened together?

Options

A. 12
B. 15
C. 20
D. 10

Answer

To solve the problem of how long it would take for taps A, B, and C to fill a tank together, we can follow these steps:

### Step 1: Determine the filling rates of each tap
- Tap A can fill the tank in 50 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{50 \text{ hours}} = \frac{1}{50} \text{ tanks per hour} \]

- Tap B can fill the tank in 60 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{60 \text{ hours}} = \frac{1}{60} \text{ tanks per hour} \]

- Tap C can fill the tank in 75 hours. Therefore, its rate is:
\[ \text{Rate of C} = \frac{1 \text{ tank}}{75 \text{ hours}} = \frac{1}{75} \text{ tanks per hour} \]

### Step 2: Calculate the combined filling rate of all taps
To find the total filling rate when all taps are opened together, we sum their individual rates:
\[ \text{Total Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the rates we calculated:
\[ \text{Total Rate} = \frac{1}{50} + \frac{1}{60} + \frac{1}{75} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 50, 60, and 75 is 300. We can convert each rate to have a denominator of 300:
\[ \frac{1}{50} = \frac{6}{300}, \quad \frac{1}{60} = \frac{5}{300}, \quad \frac{1}{75} = \frac{4}{300} \]
Now, we can add these fractions:
\[ \text{Total Rate} = \frac{6}{300} + \frac{5}{300} + \frac{4}{300} = \frac{15}{300} \]

### Step 4: Simplify the total rate
\[ \text{Total Rate} = \frac{15}{300} = \frac{1}{20} \text{ tanks per hour} \]

### Step 5: Calculate the time to fill the tank
If the combined rate is \(\frac{1}{20}\) tanks per hour, then the time taken to fill one tank is the reciprocal of the rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{20} \text{ tanks per hour}} = 20 \text{ hours} \]

### Final Answer
The three taps A, B, and C together will take **20 hours** to fill the tank.

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