Pipes A,B and C can fill a tank in 15, 20 and 30 hours respectively The first pipe A was opened at `6:00` am, B at `8:00` am, C at `9:00` am . When will the cistern be full?

Options

A. `4:18` pm
B. `2:00` pm
C. `12:15` pm
D. `11:09` am

Answer

To solve the problem of when the tank will be full, we will follow these steps:

### Step 1: Determine the rates of work for each pipe.
- Pipe A can fill the tank in 15 hours, so its rate of work is:
\[ \text{Rate of A} = \frac{1}{15} \text{ tank/hour} \]
- Pipe B can fill the tank in 20 hours, so its rate of work is:
\[ \text{Rate of B} = \frac{1}{20} \text{ tank/hour} \]
- Pipe C can fill the tank in 30 hours, so its rate of work is:
\[ \text{Rate of C} = \frac{1}{30} \text{ tank/hour} \]

### Step 2: Calculate the total work done by each pipe.
To find a common unit of work, we can use the least common multiple (LCM) of the times taken by the pipes:
- The LCM of 15, 20, and 30 is 60. This means the total work to fill the tank is 60 units.

Now, we can find the work done by each pipe in one hour:
- Work done by A in one hour:
\[ \text{Work by A} = \frac{60}{15} = 4 \text{ units/hour} \]
- Work done by B in one hour:
\[ \text{Work by B} = \frac{60}{20} = 3 \text{ units/hour} \]
- Work done by C in one hour:
\[ \text{Work by C} = \frac{60}{30} = 2 \text{ units/hour} \]

### Step 3: Calculate the work done by each pipe at different times.
- Pipe A starts working at 6:00 AM and works until 9:00 AM (3 hours):
\[ \text{Work done by A} = 4 \text{ units/hour} \times 3 \text{ hours} = 12 \text{ units} \]

- Pipe B starts working at 8:00 AM and will work until the tank is full. By 9:00 AM, A has done 12 units, so the remaining work is:
\[ 60 - 12 = 48 \text{ units} \]
B works from 8:00 AM to 9:00 AM (1 hour):
\[ \text{Work done by B} = 3 \text{ units/hour} \times 1 \text{ hour} = 3 \text{ units} \]
After 9:00 AM, the remaining work is:
\[ 48 - 3 = 45 \text{ units} \]

- Pipe C starts working at 9:00 AM. Now, A, B, and C are all working together from 9:00 AM onwards. Their combined work rate is:
\[ \text{Combined rate} = 4 + 3 + 2 = 9 \text{ units/hour} \]

### Step 4: Calculate the time required to complete the remaining work.
To find out how long it will take to complete the remaining 45 units of work:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Combined rate}} = \frac{45 \text{ units}}{9 \text{ units/hour}} = 5 \text{ hours} \]

### Step 5: Determine the final time when the tank will be full.
Since C started working at 9:00 AM and it takes 5 more hours to fill the tank:
\[ \text{Final time} = 9:00 \text{ AM} + 5 \text{ hours} = 2:00 \text{ PM} \]

Thus, the tank will be full at **2:00 PM**.

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