If three taps are opened together, a tank is filled in 12 hours. One of the taps can fill it in 10 hours and another in 15 hours. How does the third tap work ?

Options

A. empties in 12 hours
B. empties in 14 hours
C. fills in 12 hours
D. fills in 14 hours

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We have three taps (let's call them Tap X, Tap Y, and Tap Z). When all three taps are opened together, they fill the tank in 12 hours. Tap X can fill the tank in 10 hours, and Tap Y can fill it in 15 hours. We need to find out how long it takes for Tap Z to empty the tank.

### Step 2: Calculate the Filling Rates of Each Tap
1. **Tap X's Rate**:
- Tap X fills the tank in 10 hours.
- Therefore, the rate of Tap X = 1 tank / 10 hours = 1/10 tanks per hour.

2. **Tap Y's Rate**:
- Tap Y fills the tank in 15 hours.
- Therefore, the rate of Tap Y = 1 tank / 15 hours = 1/15 tanks per hour.

3. **Combined Rate of All Three Taps**:
- When all three taps are opened together, they fill the tank in 12 hours.
- Therefore, the combined rate of Tap X, Y, and Z = 1 tank / 12 hours = 1/12 tanks per hour.

### Step 3: Set Up the Equation
Now we can set up the equation based on the rates:
\[ \text{Rate of Tap X} + \text{Rate of Tap Y} + \text{Rate of Tap Z} = \text{Combined Rate} \]
\[ \frac{1}{10} + \frac{1}{15} + \text{Rate of Tap Z} = \frac{1}{12} \]

### Step 4: Find a Common Denominator
To solve this equation, we need to find a common denominator for 10, 15, and 12. The least common multiple (LCM) of these numbers is 60.

### Step 5: Convert Each Rate to the Common Denominator
1. **Tap X's Rate**:
\[ \frac{1}{10} = \frac{6}{60} \]
2. **Tap Y's Rate**:
\[ \frac{1}{15} = \frac{4}{60} \]
3. **Combined Rate**:
\[ \frac{1}{12} = \frac{5}{60} \]

### Step 6: Substitute and Solve for Tap Z's Rate
Now substituting these values into the equation:
\[ \frac{6}{60} + \frac{4}{60} + \text{Rate of Tap Z} = \frac{5}{60} \]
Combining the rates of Tap X and Tap Y:
\[ \frac{10}{60} + \text{Rate of Tap Z} = \frac{5}{60} \]
Now, isolate the rate of Tap Z:
\[ \text{Rate of Tap Z} = \frac{5}{60} - \frac{10}{60} = -\frac{5}{60} \]

### Step 7: Interpret the Result
The negative sign indicates that Tap Z is emptying the tank rather than filling it. The rate of Tap Z is:
\[ \text{Rate of Tap Z} = -\frac{5}{60} = -\frac{1}{12} \text{ tanks per hour} \]

### Step 8: Calculate the Time Taken by Tap Z to Empty the Tank
To find out how long it takes for Tap Z to empty the tank:
\[ \text{Time} = \frac{1 \text{ tank}}{\text{Rate of Tap Z}} = \frac{1}{-\frac{1}{12}} = -12 \text{ hours} \]
Since we are interested in the absolute value, Tap Z takes 12 hours to empty the tank.

### Final Answer
Tap Z can empty the tank in **12 hours**.

---