Two pipes P and Q can fill the tank alone in 60 and 90 hours respectively. If they are opened together, then in how many hours will the tank be filled?

Options

A. 35
B. 32
C. 36
D. 30

Answer

To solve the problem of how long it will take for pipes P and Q to fill the tank together, we can follow these steps:

### Step 1: Determine the rate of work for each pipe.
- Pipe P can fill the tank in 60 hours. Therefore, the rate of work for Pipe P is:
\[ \text{Rate of P} = \frac{1 \text{ tank}}{60 \text{ hours}} = \frac{1}{60} \text{ tanks per hour} \]

- Pipe Q can fill the tank in 90 hours. Therefore, the rate of work for Pipe Q is:
\[ \text{Rate of Q} = \frac{1 \text{ tank}}{90 \text{ hours}} = \frac{1}{90} \text{ tanks per hour} \]

### Step 2: Add the rates of work for both pipes.
To find the combined rate of work when both pipes are opened together, we add their individual rates:
\[ \text{Combined Rate} = \text{Rate of P} + \text{Rate of Q} = \frac{1}{60} + \frac{1}{90} \]

### Step 3: Find a common denominator and add the fractions.
The least common multiple (LCM) of 60 and 90 is 180. We convert each rate to have a denominator of 180:
\[ \frac{1}{60} = \frac{3}{180} \quad \text{and} \quad \frac{1}{90} = \frac{2}{180} \]
Now we can add the two fractions:
\[ \text{Combined Rate} = \frac{3}{180} + \frac{2}{180} = \frac{5}{180} = \frac{1}{36} \text{ tanks per hour} \]

### Step 4: Calculate the time taken to fill the tank.
If the combined rate is \(\frac{1}{36}\) tanks per hour, then the time taken to fill 1 tank is the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{36} \text{ tanks per hour}} = 36 \text{ hours} \]

### Final Answer:
The tank will be filled in **36 hours**.

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