Tap A can fill a tank in 24 minutes and tap B can fill the same tank in 30 minutes, another tap C can empty the tank in 20 minutes. If tap A and tap B are opened together & after six minutes tap C is also opened, then find the total time taken to fill the remaining tank?

Options

A. A)18 minutes
B. B)14 minutes
C. C)22 minutes
D. D)12 minutes

Answer

To solve the problem step by step, we will first determine the efficiencies of taps A, B, and C, and then calculate the total time taken to fill the tank.

### Step 1: Determine the efficiencies of taps A, B, and C.
- Tap A can fill the tank in 24 minutes. Therefore, its efficiency is:
\[ \text{Efficiency of A} = \frac{1 \text{ tank}}{24 \text{ minutes}} = \frac{120}{24} = 5 \text{ units/minute} \]
- Tap B can fill the tank in 30 minutes. Therefore, its efficiency is:
\[ \text{Efficiency of B} = \frac{1 \text{ tank}}{30 \text{ minutes}} = \frac{120}{30} = 4 \text{ units/minute} \]
- Tap C can empty the tank in 20 minutes. Therefore, its efficiency (as a negative value since it empties) is:
\[ \text{Efficiency of C} = -\frac{1 \text{ tank}}{20 \text{ minutes}} = -\frac{120}{20} = -6 \text{ units/minute} \]

### Step 2: Calculate the combined efficiency of taps A and B when opened together.
- The combined efficiency of taps A and B is:
\[ \text{Combined efficiency of A and B} = 5 + 4 = 9 \text{ units/minute} \]

### Step 3: Calculate the amount of work done by taps A and B in the first 6 minutes.
- In 6 minutes, the amount of work done by taps A and B is:
\[ \text{Work done in 6 minutes} = \text{Efficiency} \times \text{Time} = 9 \text{ units/minute} \times 6 \text{ minutes} = 54 \text{ units} \]

### Step 4: Calculate the remaining work to be done.
- The total work required to fill the tank is 120 units. After 6 minutes, the remaining work is:
\[ \text{Remaining work} = 120 - 54 = 66 \text{ units} \]

### Step 5: Calculate the combined efficiency when tap C is also opened.
- When tap C is opened along with taps A and B, the combined efficiency becomes:
\[ \text{Combined efficiency of A, B, and C} = 5 + 4 - 6 = 3 \text{ units/minute} \]

### Step 6: Calculate the time taken to fill the remaining work with all three taps open.
- Let \( T \) be the time taken to fill the remaining 66 units of work. We can set up the equation:
\[ \text{Efficiency} \times \text{Time} = \text{Remaining work} \]
\[ 3 \text{ units/minute} \times T = 66 \text{ units} \]
\[ T = \frac{66}{3} = 22 \text{ minutes} \]

### Final Answer:
The total time taken to fill the remaining tank after tap C is opened is **22 minutes**.

---