Two pipes A and B can fill a tank in 24h and 30 h respectively. If both the pipes are opened simultaneously in the empty tank, how much time will be taken by them to fill it?

### Concept & Formula When two pipes work together, their combined filling rate is the sum of their individual filling rates. $$\text{Combined Rate} = \frac{1}{A} + \frac{1}{B}$$ $$\text{Total Time} = \frac{A \times B}{A + B}$$ ### Step-by-Step Solution 1. Pipe A fills the tank in 24 hours. Its rate is $\frac{1}{24}$ per hour. 2. Pipe B fills the tank in 30 hours. Its rate is $\frac{1}{30}$ per hour. 3. Combined rate when both are open: $$\frac{1}{24} + \frac{1}{30}$$ 4. Find a common denominator (LCM of 24 and 30 is 120): $$\text{Combined Rate} = \frac{5}{120} + \frac{4}{120} = \frac{9}{120} = \frac{3}{40} \text{ tank per hour}$$ 5. Total time taken is the reciprocal of the combined rate: $$\text{Time} = \frac{40}{3} \text{ hours} = 13 \frac{1}{3} \text{ hours}$$ 6. Convert the fractional hour to minutes: $$\frac{1}{3} \text{ hour} = \frac{1}{3} \times 60 \text{ minutes} = 20 \text{ minutes}$$ Total time = 13h 20 min. ### Exam Strategy & Shortcut Use the direct formula $T = \frac{ab}{a+b}$. $$T = \frac{24 \times 30}{24 + 30} = \frac{720}{54}$$ Simplify by dividing by 18: $$T = \frac{40}{3} = 13.33 \text{ hours}$$, which is 13 hours and 20 minutes. ### Common Pitfall A frequent error is adding the hours directly ($24 + 30 = 54$) instead of adding their rates (inverses). Always add rates, never times. ### Final Answer Therefore, the correct answer is **13h 20 min**.