Pipes A, B, and C together can fill a tank in 45 minutes, while pipes B and C together take 75 minutes to fill the same tank. Given that pipe B is 100\% more efficient than pipe C, find the time taken by pipes A and B together to fill the tank.

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**Solution:**

1. Let the efficiency (rate of filling) of pipe C be *x* units per minute.

2. Since pipe B is 100\% more efficient than pipe C, efficiency of pipe B = *2x* units per minute.

3. Efficiency of pipes B and C together = *x + 2x = 3x* units per minute.

4. Given that pipes B and C together fill the tank in 75 minutes, so:
   $$	ext{Rate of B and C} = \frac{1}{75} = 3x \implies x = \frac{1}{225}$$

5. Therefore, efficiency of pipe B = $2x = \frac{2}{225}$ and pipe C = $\frac{1}{225}$ units per minute.

6. Pipes A, B, and C together fill the tank in 45 minutes, so their combined rate:
   $$\frac{1}{45} = 	ext{Efficiency of A} + 	ext{Efficiency of B} + 	ext{Efficiency of C}$$

7. Substitute values for B and C:
   $$\frac{1}{45} = 	ext{Efficiency of A} + \frac{2}{225} + \frac{1}{225} = 	ext{Efficiency of A} + \frac{3}{225}$$

8. Simplify $\frac{3}{225} = \frac{1}{75}$, so:
   $$	ext{Efficiency of A} = \frac{1}{45} - \frac{1}{75} = \frac{5 - 3}{225} = \frac{2}{225}$$

9. Efficiency of A and B together:
   $$\frac{2}{225} + \frac{2}{225} = \frac{4}{225}$$

10. Time taken by A and B together to fill the tank:
    $$	ext{Time} = \frac{1}{	ext{Efficiency}} = \frac{1}{\frac{4}{225}} = \frac{225}{4} = 56.25 	ext{ minutes}$$

**Answer:** 56.25 minutes

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