Two inlet taps A and B can fill a tank in 36 minutes and 60 minutes respectively. Find the time taken by both the taps together to fill `(1)/(6)`th of the tank?

Options

A. 3 minutes
B. `3(3)/(4)` minutes
C. `3(1)/(2)` minutes
D. `3(1)/(3)` minutes

Answer

To solve the problem of how long it takes for both taps A and B to fill \( \frac{1}{6} \) of the tank together, we can follow these steps:

### Step 1: Determine the individual efficiencies of taps A and B.
- Tap A can fill the tank in 36 minutes, so its efficiency is:
\[ \text{Efficiency of A} = \frac{1 \text{ tank}}{36 \text{ minutes}} = \frac{1}{36} \text{ tanks per minute} \]
- Tap B can fill the tank in 60 minutes, so its efficiency is:
\[ \text{Efficiency of B} = \frac{1 \text{ tank}}{60 \text{ minutes}} = \frac{1}{60} \text{ tanks per minute} \]

### Step 2: Calculate the combined efficiency of taps A and B.
- The combined efficiency when both taps are open is:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{36} + \frac{1}{60} \]
- To add these fractions, we need a common denominator. The least common multiple of 36 and 60 is 180.
\[ \frac{1}{36} = \frac{5}{180}, \quad \frac{1}{60} = \frac{3}{180} \]
- Therefore,
\[ \text{Combined Efficiency} = \frac{5}{180} + \frac{3}{180} = \frac{8}{180} = \frac{2}{45} \text{ tanks per minute} \]

### Step 3: Determine the amount of work needed to fill \( \frac{1}{6} \) of the tank.
- The total work to fill \( \frac{1}{6} \) of the tank is:
\[ \text{Work for } \frac{1}{6} \text{ tank} = \frac{1}{6} \text{ tanks} \]

### Step 4: Calculate the time taken to fill \( \frac{1}{6} \) of the tank using the combined efficiency.
- The time taken to fill \( \frac{1}{6} \) of the tank is given by:
\[ \text{Time} = \frac{\text{Work}}{\text{Combined Efficiency}} = \frac{\frac{1}{6}}{\frac{2}{45}} = \frac{1}{6} \times \frac{45}{2} = \frac{45}{12} = \frac{15}{4} \text{ minutes} \]

### Step 5: Convert the time into minutes and seconds.
- \( \frac{15}{4} \) minutes can be expressed as:
\[ 3 \text{ minutes } + \frac{3}{4} \text{ minutes} = 3 \text{ minutes } + 45 \text{ seconds} \]
- Thus, the time taken by both taps together to fill \( \frac{1}{6} \) of the tank is:
\[ 3 \text{ minutes } 45 \text{ seconds} \text{ or } 3 \frac{3}{4} \text{ minutes} \]

### Final Answer:
The time taken by both taps A and B together to fill \( \frac{1}{6} \) of the tank is \( 3 \frac{3}{4} \) minutes.

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