In what time can pipe C alone fill the tank?

Statement I: Pipe A and B together take 4 hours to fill the tank, while pipes A, B, and C together take 3 hours to fill the tank. All pipes are inlet pipes.

Statement II: The time taken by pipe A alone to fill half of the tank is the same as the time taken by pipes A and B together to fill the entire tank.

Question

In what time can pipe C alone fill the tank?

Statement I: Pipe A and B together take 4 hours to fill the tank, while pipes A, B, and C together take 3 hours to fill the tank. All pipes are inlet pipes.

Statement II: The time taken by pipe A alone to fill half of the tank is the same as the time taken by pipes A and B together to fill the entire tank.

Accepted Answer

**Solution:**

1. Let the rates of pipes A, B, and C be *a*, *b*, and *c* tanks per hour respectively.

2. From Statement I:
   - Pipes A and B together take 4 hours to fill the tank, so their combined rate is:
     $$a + b = \frac{1}{4} 	ext{ tank/hour}$$
   - Pipes A, B, and C together take 3 hours to fill the tank, so their combined rate is:
     $$a + b + c = \frac{1}{3} 	ext{ tank/hour}$$

3. Using these two equations, we can find the rate of pipe C:
   $$c = \frac{1}{3} - \frac{1}{4} = \frac{4 - 3}{12} = \frac{1}{12} 	ext{ tank/hour}$$

So, pipe C alone can fill the tank in 12 hours.

4. From Statement II:
   - Time taken by pipe A alone to fill half the tank equals the time taken by pipes A and B together to fill the whole tank.
   - Time by A alone to fill half tank = Time by A and B together to fill full tank.
   - Let time taken by A alone to fill half tank be *t* hours.
   - Then, rate of A is:
     $$a = \frac{0.5}{t} = \frac{1}{2t}$$
   - Rate of A and B together is:
     $$a + b = \frac{1}{t}$$

- From above, we get:
     $$a + b = 2a$$
     $$\Rightarrow b = a$$

- This gives a relation between *a* and *b* but does not provide their exact values or *c*.

5. Statement II alone is insufficient to find pipe C's time.

6. Statement I alone allows calculation of pipe C's rate and time.

7. However, to confirm the rates and ensure consistency, both statements together are necessary.

**Answer:** Both statements together are necessary to answer.

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