If 3 taps are open together, a cistern is filled in 3 hrs. One of the taps alone can fill it in 10 hrs and another in 15 hrs. In how many hours does the third tap fill it?

Answer

To solve the problem, we will use the concept of rates of work done by each tap. Let's denote the time taken by the third tap to fill the cistern as \( x \) hours.

### Step 1: Determine the rates of work for each tap
1. The first tap can fill the cistern in 10 hours. Therefore, its rate of work is:
\[ \text{Rate of Tap 1} = \frac{1}{10} \text{ cisterns per hour} \]

2. The second tap can fill the cistern in 15 hours. Therefore, its rate of work is:
\[ \text{Rate of Tap 2} = \frac{1}{15} \text{ cisterns per hour} \]

3. The third tap can fill the cistern in \( x \) hours. Therefore, its rate of work is:
\[ \text{Rate of Tap 3} = \frac{1}{x} \text{ cisterns per hour} \]

### Step 2: Set up the equation for combined rates
When all three taps are open together, they fill the cistern in 3 hours. Thus, their combined rate of work is:
\[ \text{Combined Rate} = \frac{1}{3} \text{ cisterns per hour} \]

### Step 3: Write the equation
The combined rate of the three taps can be expressed as:
\[ \frac{1}{10} + \frac{1}{15} + \frac{1}{x} = \frac{1}{3} \]

### Step 4: Find a common denominator
The least common multiple (LCM) of 10, 15, and 3 is 30. We will use this to eliminate the fractions:
\[ \frac{3}{30} + \frac{2}{30} + \frac{30}{x} = \frac{10}{30} \]

### Step 5: Simplify the equation
Combining the fractions on the left side gives:
\[ \frac{5}{30} + \frac{30}{x} = \frac{10}{30} \]

### Step 6: Isolate the term with \( x \)
Subtract \( \frac{5}{30} \) from both sides:
\[ \frac{30}{x} = \frac{10}{30} - \frac{5}{30} \]
\[ \frac{30}{x} = \frac{5}{30} \]

### Step 7: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 30 \cdot 30 = 5 \cdot x \]
\[ 900 = 5x \]

### Step 8: Solve for \( x \)
Dividing both sides by 5:
\[ x = \frac{900}{5} = 180 \]

### Conclusion
The third tap can fill the cistern in **180 hours**.

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