Three taps P Q and R can fill a tank in 20, 30 and 40 minutes respectively. If all the three taps are opened, then how much time (in minutes) it will take to completely fill the tank?

Options

A. `9(2)/(13)`
B. 12
C. `8(1)/(3)`
D. 15

Answer

To solve the problem of how long it will take to fill the tank when all three taps P, Q, and R are opened, we can follow these steps:

### Step 1: Determine the filling rates of each tap
- Tap P can fill the tank in 20 minutes, so its rate is \( \frac{1}{20} \) of the tank per minute.
- Tap Q can fill the tank in 30 minutes, so its rate is \( \frac{1}{30} \) of the tank per minute.
- Tap R can fill the tank in 40 minutes, so its rate is \( \frac{1}{40} \) of the tank per minute.

### Step 2: Calculate the combined filling rate of all taps
To find the total rate at which the tank is filled when all taps are opened, we need to add the rates of the three taps:

\[ \text{Combined rate} = \frac{1}{20} + \frac{1}{30} + \frac{1}{40} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 20, 30, and 40 is 120. We will convert each rate to have this common denominator:

\[ \frac{1}{20} = \frac{6}{120}, \quad \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120} \]

### Step 4: Add the rates
Now, we can add the rates:

\[ \text{Combined rate} = \frac{6}{120} + \frac{4}{120} + \frac{3}{120} = \frac{13}{120} \]

This means that together, the taps fill \( \frac{13}{120} \) of the tank in one minute.

### Step 5: Calculate the time to fill the tank
To find out how long it will take to fill the entire tank (1 tank), we take the reciprocal of the combined rate:

\[ \text{Time} = \frac{1 \text{ tank}}{\frac{13}{120} \text{ tanks/minute}} = \frac{120}{13} \text{ minutes} \]

Calculating \( \frac{120}{13} \):

\[ \frac{120}{13} \approx 9.23 \text{ minutes} \]

Thus, it will take approximately 9.23 minutes to fill the tank when all three taps are opened.

### Final Answer
The time taken to completely fill the tank when all taps are opened is approximately **9.23 minutes**.

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