Three taps A ,B and C together can fill an empty cistern in 10 min . The tap A alone can fill it in 30 min and the tapBalone can fill it in 40 min. How long will the tap C alone take to fill it?

Options

A. 16min
B. 24min
C. 32min
D. 40min

Answer

To solve the problem step by step, we will first determine the rates at which each tap fills the cistern and then find out how long tap C will take to fill it alone.

### Step 1: Determine the filling rates of taps A, B, and C

1. **Tap A's rate**: Tap A can fill the cistern in 30 minutes.
- Rate of A = 1 cistern / 30 minutes = \( \frac{1}{30} \) cisterns per minute.

2. **Tap B's rate**: Tap B can fill the cistern in 40 minutes.
- Rate of B = 1 cistern / 40 minutes = \( \frac{1}{40} \) cisterns per minute.

3. **Combined rate of A, B, and C**: Together, taps A, B, and C can fill the cistern in 10 minutes.
- Rate of A + B + C = 1 cistern / 10 minutes = \( \frac{1}{10} \) cisterns per minute.

### Step 2: Set up the equation

Now, we can express the combined rate of A, B, and C in terms of their individual rates:

\[ \text{Rate of A} + \text{Rate of B} + \text{Rate of C} = \text{Rate of A + B + C} \]

Substituting the known rates:

\[ \frac{1}{30} + \frac{1}{40} + \text{Rate of C} = \frac{1}{10} \]

### Step 3: Find a common denominator and simplify

To solve for Rate of C, we need a common denominator for the fractions. The least common multiple (LCM) of 30, 40, and 10 is 120.

- Convert each rate:
- \( \frac{1}{30} = \frac{4}{120} \)
- \( \frac{1}{40} = \frac{3}{120} \)
- \( \frac{1}{10} = \frac{12}{120} \)

Now substitute these values into the equation:

\[ \frac{4}{120} + \frac{3}{120} + \text{Rate of C} = \frac{12}{120} \]

### Step 4: Solve for Rate of C

Combine the rates of A and B:

\[ \frac{4 + 3}{120} + \text{Rate of C} = \frac{12}{120} \]

This simplifies to:

\[ \frac{7}{120} + \text{Rate of C} = \frac{12}{120} \]

Now, isolate Rate of C:

\[ \text{Rate of C} = \frac{12}{120} - \frac{7}{120} = \frac{5}{120} = \frac{1}{24} \]

### Step 5: Determine the time taken by tap C

Since Rate of C = \( \frac{1}{24} \) cisterns per minute, it means tap C can fill the cistern in 24 minutes.

### Final Answer

Tap C alone will take **24 minutes** to fill the cistern.

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