Two pipes P and Q can fill an empty tank in 25 hours and 20 hours respectively. Pipe R alone can empty the completely filled tank in 50 hours. Firstly both the pipes P and Q are opened and after 8 hours, pipe R is also opened. What will be the total time (in hours) taken to completely fill the tank?

Options

A. 12
B. 10
C. 14
D. 13

Answer

To solve the problem step by step, we will calculate the filling and emptying rates of the pipes, determine how much of the tank is filled after 8 hours, and then find out how long it takes to fill the remaining part of the tank after pipe R is opened.

### Step 1: Determine the filling rates of pipes P and Q
- Pipe P can fill the tank in 25 hours.
- Pipe Q can fill the tank in 20 hours.

To find the rate at which each pipe fills the tank:
- Rate of Pipe P = 1 tank / 25 hours = 1/25 tanks per hour
- Rate of Pipe Q = 1 tank / 20 hours = 1/20 tanks per hour

### Step 2: Calculate the combined filling rate of pipes P and Q
- Combined rate of P and Q = Rate of P + Rate of Q
- Combined rate = (1/25 + 1/20)

To add these fractions, we need a common denominator:
- The least common multiple of 25 and 20 is 100.
- Convert the rates:
- (1/25) = 4/100
- (1/20) = 5/100

Now, add the rates:
- Combined rate = 4/100 + 5/100 = 9/100 tanks per hour

### Step 3: Calculate the amount of tank filled in the first 8 hours
- Amount filled in 8 hours = Combined rate × Time
- Amount filled = (9/100) × 8 = 72/100 = 0.72 tanks

### Step 4: Determine the emptying rate of pipe R
- Pipe R can empty the tank in 50 hours.
- Rate of Pipe R = 1 tank / 50 hours = 1/50 tanks per hour

### Step 5: Calculate the net rate when all pipes are opened
- When pipes P, Q, and R are opened together:
- Net rate = Combined rate of P and Q - Rate of R
- Net rate = (9/100) - (1/50)

Convert (1/50) to a fraction with a denominator of 100:
- (1/50) = 2/100

Now, calculate the net rate:
- Net rate = (9/100) - (2/100) = 7/100 tanks per hour

### Step 6: Calculate the remaining amount of the tank to be filled
- Total tank capacity = 1 tank
- Amount already filled = 0.72 tanks
- Remaining amount = 1 - 0.72 = 0.28 tanks

### Step 7: Calculate the time taken to fill the remaining amount with all pipes open
- Time = Remaining amount / Net rate
- Time = 0.28 / (7/100) = 0.28 × (100/7) = 4 hours

### Step 8: Calculate the total time taken to fill the tank
- Total time = Time with P and Q + Time with P, Q, and R
- Total time = 8 hours + 4 hours = 12 hours

### Final Answer
The total time taken to completely fill the tank is **12 hours**.

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