A tank can be filled by two taps P and Q in 15 hours and 20 hours respectively. The full tank can be emptied by a third tap R in 10 hours. If all the three taps are turned on at the same time, then in how much time will the empty tank be filled up completely?

Options

A. 30 hours
B. 45 hours
C. 40 hours
D. 60 hours

Answer

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

### Step 1: Determine the filling rates of taps P and Q, and the emptying rate of tap R.

- Tap P fills the tank in 15 hours. Therefore, the rate of tap P is:
\[ \text{Rate of P} = \frac{1}{15} \text{ tank per hour} \]

- Tap Q fills the tank in 20 hours. Therefore, the rate of tap Q is:
\[ \text{Rate of Q} = \frac{1}{20} \text{ tank per hour} \]

- Tap R empties the tank in 10 hours. Therefore, the rate of tap R is:
\[ \text{Rate of R} = -\frac{1}{10} \text{ tank per hour} \quad (\text{negative because it empties}) \]

### Step 2: Combine the rates of all three taps.

When all three taps are opened, the combined rate of filling the tank is:
\[ \text{Combined Rate} = \text{Rate of P} + \text{Rate of Q} + \text{Rate of R} \]
Substituting the values we found:
\[ \text{Combined Rate} = \frac{1}{15} + \frac{1}{20} - \frac{1}{10} \]

### Step 3: Find a common denominator and simplify.

The least common multiple (LCM) of 15, 20, and 10 is 60. Now we convert each fraction:
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{10} = \frac{6}{60} \]
Now substituting these values into the combined rate:
\[ \text{Combined Rate} = \frac{4}{60} + \frac{3}{60} - \frac{6}{60} = \frac{4 + 3 - 6}{60} = \frac{1}{60} \text{ tank per hour} \]

### Step 4: Calculate the time to fill the tank.

If the combined rate is \(\frac{1}{60}\) tank per hour, then the time \(T\) to fill 1 tank is given by:
\[ T = \frac{1 \text{ tank}}{\text{Combined Rate}} = \frac{1}{\frac{1}{60}} = 60 \text{ hours} \]

### Final Answer:
Thus, the time taken to fill the tank when all three taps are opened is **60 hours**.

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