Two pipe A and B can fill a tank with water in 42min and 35min respectively. The third pipe C can empty the tank in 60min. First A and B are opened. After 12 minutes C is opened. Total time (in min) in which the tank will be filled up is?

Options

A. 22 min 24 sec
B. 23 min 36 sec
C. 21 min 27 sec
D. 24 min 40 sec

Answer

To solve the problem step by step, we will first determine the filling rates of pipes A, B, and C, then calculate how much water is filled in the first 12 minutes, and finally find out how long it takes to fill the rest of the tank after pipe C is opened.

### Step 1: Determine the filling rates of pipes A, B, and C.
- Pipe A fills the tank in 42 minutes.
- Pipe B fills the tank in 35 minutes.
- Pipe C empties the tank in 60 minutes.

**Filling rates:**
- Rate of A = \( \frac{1}{42} \) tank/minute
- Rate of B = \( \frac{1}{35} \) tank/minute
- Rate of C = \( -\frac{1}{60} \) tank/minute (negative because it empties)

### Step 2: Calculate the combined filling rate of A and B.
- Combined rate of A and B = Rate of A + Rate of B
- Combined rate = \( \frac{1}{42} + \frac{1}{35} \)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 42 and 35 is 210.

- Rate of A = \( \frac{5}{210} \)
- Rate of B = \( \frac{6}{210} \)

So, combined rate of A and B:
\[ \frac{5}{210} + \frac{6}{210} = \frac{11}{210} \text{ tank/minute} \]

### Step 3: Calculate the amount of water filled by A and B in 12 minutes.
- Amount filled in 12 minutes = Combined rate × Time
- Amount filled = \( \frac{11}{210} \times 12 = \frac{132}{210} = \frac{22}{35} \text{ tank} \)

### Step 4: Determine the remaining amount of the tank to be filled.
- Total tank = 1 tank
- Remaining amount = Total tank - Amount filled
- Remaining amount = \( 1 - \frac{22}{35} = \frac{13}{35} \text{ tank} \)

### Step 5: Calculate the new combined rate when pipe C is opened.
- Combined rate of A, B, and C = Combined rate of A and B + Rate of C
- Combined rate = \( \frac{11}{210} - \frac{1}{60} \)

To add these fractions, we need a common denominator. The LCM of 210 and 60 is 630.

- Rate of A and B = \( \frac{11 \times 3}{630} = \frac{33}{630} \)
- Rate of C = \( -\frac{1 \times 10.5}{630} = -\frac{10.5}{630} \)

So, combined rate of A, B, and C:
\[ \frac{33}{630} - \frac{10.5}{630} = \frac{22.5}{630} = \frac{15}{420} \text{ tank/minute} \]

### Step 6: Calculate the time required to fill the remaining amount with all three pipes open.
- Time = Remaining amount ÷ Combined rate
- Time = \( \frac{13/35}{15/420} = \frac{13 \times 420}{35 \times 15} \)

Calculating this gives:
- Time = \( \frac{5460}{525} = 10.4 \text{ minutes} \)

### Step 7: Calculate the total time taken to fill the tank.
- Total time = Time with A and B + Time with A, B, and C
- Total time = \( 12 + 10.4 = 22.4 \text{ minutes} \)

### Final Answer:
The total time taken to fill the tank is **22 minutes and 24 seconds**.

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