Pipes A and B can fill a tank in 15 hours and 12 ,respectively .Pipe C alone can empty the full tank in 10 hours .If all the three pipes are opened together for 2 hours 40 minutes ,then what part of the tank wil remain unfilled ?

Options

A. `(3)/(20)`
B. `(13)/(15)`
C. `(2)/(15)`
D. `(17)/(20)`

Answer

To solve the problem step by step, we need to determine how much of the tank remains unfilled after all three pipes are opened together for 2 hours and 40 minutes.

### Step 1: Determine the filling and emptying rates of the pipes.

- **Pipe A** fills the tank in 15 hours. Therefore, its rate of filling is:
\[ \text{Rate of A} = \frac{1}{15} \text{ tank/hour} \]

- **Pipe B** fills the tank in 12 hours. Therefore, its rate of filling is:
\[ \text{Rate of B} = \frac{1}{12} \text{ tank/hour} \]

- **Pipe C** empties the tank in 10 hours. Therefore, its rate of emptying is:
\[ \text{Rate of C} = -\frac{1}{10} \text{ tank/hour} \]

### Step 2: Calculate the combined rate of all three pipes.

To find the combined rate when all three pipes are opened, we add the rates of A and B and subtract the rate of C:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
\[ = \frac{1}{15} + \frac{1}{12} - \frac{1}{10} \]

### Step 3: Find a common denominator and compute the combined rate.

The least common multiple (LCM) of 15, 12, and 10 is 60. Now we convert each rate to have the same denominator:
\[ \text{Rate of A} = \frac{4}{60}, \quad \text{Rate of B} = \frac{5}{60}, \quad \text{Rate of C} = -\frac{6}{60} \]
Now, adding these rates together:
\[ \text{Combined Rate} = \frac{4}{60} + \frac{5}{60} - \frac{6}{60} = \frac{3}{60} = \frac{1}{20} \text{ tank/hour} \]

### Step 4: Calculate the total time in hours.

The total time for which the pipes are open is 2 hours and 40 minutes. We convert this to hours:
\[ 2 \text{ hours} + \frac{40 \text{ minutes}}{60} = 2 + \frac{2}{3} = \frac{8}{3} \text{ hours} \]

### Step 5: Calculate the amount of the tank filled in that time.

Using the combined rate, we can find out how much of the tank is filled in \(\frac{8}{3}\) hours:
\[ \text{Amount filled} = \text{Combined Rate} \times \text{Time} = \frac{1}{20} \times \frac{8}{3} = \frac{8}{60} = \frac{2}{15} \text{ of the tank} \]

### Step 6: Determine the part of the tank that remains unfilled.

Since the total capacity of the tank is 1, the part of the tank that remains unfilled is:
\[ \text{Unfilled part} = 1 - \text{Amount filled} = 1 - \frac{2}{15} = \frac{15}{15} - \frac{2}{15} = \frac{13}{15} \]

### Final Answer:
The part of the tank that will remain unfilled is \(\frac{13}{15}\).

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