Tap A can fill a tank in 6 hours and tap B can empty the same tank in 10 hours. If both taps are opened together, then how much time (in hours) will be taken to fill the rank?

Options

A. 16
B. 15
C. 18
D. 20

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the filling rate of Tap A
Tap A can fill the tank in 6 hours. Therefore, in one hour, Tap A can fill:
\[ \text{Filling rate of Tap A} = \frac{1}{6} \text{ tank/hour} \]

### Step 2: Determine the emptying rate of Tap B
Tap B can empty the tank in 10 hours. Therefore, in one hour, Tap B can empty:
\[ \text{Emptying rate of Tap B} = \frac{1}{10} \text{ tank/hour} \]

### Step 3: Calculate the net rate when both taps are opened
When both taps are opened together, the net effect on the tank is the filling rate of Tap A minus the emptying rate of Tap B:
\[ \text{Net rate} = \text{Filling rate of Tap A} - \text{Emptying rate of Tap B} = \frac{1}{6} - \frac{1}{10} \]

### Step 4: Find a common denominator and simplify
To subtract these fractions, we need a common denominator. The least common multiple of 6 and 10 is 30. Thus, we convert the fractions:
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{10} = \frac{3}{30} \]
Now we can subtract:
\[ \text{Net rate} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15} \text{ tank/hour} \]

### Step 5: Calculate the time to fill the tank
Since the net rate of filling the tank is \(\frac{1}{15}\) tank/hour, it means that the time taken to fill the entire tank is the reciprocal of the net rate:
\[ \text{Time to fill the tank} = \frac{1}{\text{Net rate}} = \frac{1}{\frac{1}{15}} = 15 \text{ hours} \]

### Final Answer
The time taken to fill the tank when both taps are opened together is **15 hours**.

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