A tank can be filled by pipe A in 2 hours and pipe B in 6 hours. At 10 A.M. pipe A was opened. At what time will the tank be filled if pipe B is opened at 11 A.M.?

Options

A. 12.45 A.M.
B. 5 P.M.
C. 11.45 A.M.
D. 12 P.M.

Answer

To solve the problem step by step, we will calculate how much of the tank is filled by pipe A and then determine how long it takes for both pipes A and B to fill the remaining part of the tank.

### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 2 hours.
- Pipe B can fill the tank in 6 hours.

**Filling rate of Pipe A:**
\[ \text{Rate of A} = \frac{1 \text{ tank}}{2 \text{ hours}} = 0.5 \text{ tanks per hour} \]

**Filling rate of Pipe B:**
\[ \text{Rate of B} = \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tanks per hour} \]

### Step 2: Calculate the amount filled by Pipe A from 10 A.M. to 11 A.M.
From 10 A.M. to 11 A.M., only Pipe A is open for 1 hour.
\[ \text{Amount filled by A in 1 hour} = 0.5 \text{ tanks} \]

### Step 3: Determine the remaining capacity of the tank
Since the tank is filled by 0.5 tanks by Pipe A in the first hour, the remaining capacity of the tank is:
\[ \text{Remaining capacity} = 1 - 0.5 = 0.5 \text{ tanks} \]

### Step 4: Calculate the combined filling rate of Pipes A and B
From 11 A.M. onwards, both pipes A and B are open. The combined filling rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = 0.5 + \frac{1}{6} \]
To add these, we convert 0.5 to a fraction:
\[ 0.5 = \frac{3}{6} \]
Now add:
\[ \text{Combined Rate} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \text{ tanks per hour} \]

### Step 5: Calculate the time taken to fill the remaining capacity
We need to fill the remaining 0.5 tanks at a rate of \(\frac{2}{3}\) tanks per hour.
Let \(t\) be the time in hours to fill the remaining capacity:
\[ \frac{2}{3} t = 0.5 \]
Solving for \(t\):
\[ t = 0.5 \div \frac{2}{3} = 0.5 \times \frac{3}{2} = \frac{3}{4} \text{ hours} \]
Convert \(\frac{3}{4}\) hours to minutes:
\[ \frac{3}{4} \text{ hours} = 45 \text{ minutes} \]

### Step 6: Determine the final time
Since both pipes A and B start working together at 11 A.M. and it takes 45 minutes to fill the remaining capacity, the tank will be completely filled at:
\[ 11:00 \text{ A.M.} + 45 \text{ minutes} = 11:45 \text{ A.M.} \]

### Final Answer:
The tank will be completely filled at **11:45 A.M.**

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