A tap can empty a tank in 30 minutes. A second tap can empty it in 45 minutes. If both the taps operate simultaneously, how much time is needed to empty the tank?

Options

A. 30 minutes
B. 18 minutes
C. 14 minutes
D. 15 minutes

Answer

To solve the problem of how long it takes to empty a tank using two taps that operate simultaneously, we can follow these steps:

### Step 1: Determine the rates of each tap
- **Tap A** can empty the tank in 30 minutes. Therefore, in 1 minute, it can empty \( \frac{1}{30} \) of the tank.
- **Tap B** can empty the tank in 45 minutes. Therefore, in 1 minute, it can empty \( \frac{1}{45} \) of the tank.

### Step 2: Find the combined rate of both taps
To find out how much of the tank is emptied in one minute when both taps are open, we add their individual rates:
\[ \text{Combined rate} = \frac{1}{30} + \frac{1}{45} \]

### Step 3: Calculate the least common multiple (LCM)
To add the fractions, we need a common denominator. The LCM of 30 and 45 is 90.

### Step 4: Convert the rates to the common denominator
Now, we convert the rates:
\[ \frac{1}{30} = \frac{3}{90} \quad \text{and} \quad \frac{1}{45} = \frac{2}{90} \]
So,
\[ \text{Combined rate} = \frac{3}{90} + \frac{2}{90} = \frac{5}{90} \]

### Step 5: Simplify the combined rate
The combined rate can be simplified:
\[ \frac{5}{90} = \frac{1}{18} \]
This means that together, both taps can empty \( \frac{1}{18} \) of the tank in one minute.

### Step 6: Calculate the total time to empty the tank
If both taps can empty \( \frac{1}{18} \) of the tank in one minute, then to empty the entire tank (1 whole tank), it will take:
\[ \text{Time} = 1 \div \frac{1}{18} = 18 \text{ minutes} \]

### Final Answer
Thus, the total time needed to empty the tank using both taps simultaneously is **18 minutes**.

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