Three taps R, S and T can fill a tank in 90, 100 and 180 min respectively. If all the taps are opened together, then in how many minutes will the tank be filled ?

Options

A. 120 min
B. 37.5 min
C. 45 min
D. 90 min

Answer

To solve the problem of how long it will take for taps R, S, and T to fill a tank when opened together, we can follow these steps:

### Step 1: Determine the filling rates of each tap.
- Tap R can fill the tank in 90 minutes.
- Tap S can fill the tank in 100 minutes.
- Tap T can fill the tank in 180 minutes.

### Step 2: Calculate the filling rates (efficiencies) of each tap.
- The filling rate of tap R = 1 tank / 90 minutes = \( \frac{1}{90} \) tanks per minute.
- The filling rate of tap S = 1 tank / 100 minutes = \( \frac{1}{100} \) tanks per minute.
- The filling rate of tap T = 1 tank / 180 minutes = \( \frac{1}{180} \) tanks per minute.

### Step 3: Find the total filling rate when all taps are open together.
- Total filling rate = Rate of R + Rate of S + Rate of T
- Total filling rate = \( \frac{1}{90} + \frac{1}{100} + \frac{1}{180} \)

### Step 4: Calculate the least common multiple (LCM) of the denominators (90, 100, 180) to combine the fractions.
- The LCM of 90, 100, and 180 is 900.
- Convert each fraction to have a denominator of 900:
- \( \frac{1}{90} = \frac{10}{900} \)
- \( \frac{1}{100} = \frac{9}{900} \)
- \( \frac{1}{180} = \frac{5}{900} \)

### Step 5: Add the fractions.
- Total filling rate = \( \frac{10}{900} + \frac{9}{900} + \frac{5}{900} = \frac{24}{900} \)

### Step 6: Simplify the total filling rate.
- Total filling rate = \( \frac{24}{900} = \frac{4}{150} = \frac{2}{75} \) tanks per minute.

### Step 7: Calculate the time taken to fill one tank.
- Time = Work done / Efficiency
- Work done = 1 tank
- Efficiency = \( \frac{2}{75} \) tanks per minute
- Time = \( \frac{1}{\frac{2}{75}} = \frac{75}{2} = 37.5 \) minutes.

### Final Answer:
The tank will be filled in **37.5 minutes** when all three taps are opened together.

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