Two pipes A and B can fill a tank in 23 hours and 17 hours, respectively. If…

Concept: When two or more pipes fill a tank together, their combined rate of filling is the sum of their individual rates. A pipe that alone fills a tank in t hours does \(\frac{1}{t}\) of the work per hour, and the time taken to finish the job together is the reciprocal of the sum of the individual rates.

Application:

1. Pipe A alone fills the tank in 23 hours, so its rate of work is \(\frac{1}{23}\) of the tank per hour.

2. Pipe B alone fills the tank in 17 hours, so its rate of work is \(\frac{1}{17}\) of the tank per hour.

3. When both pipes are opened simultaneously, the combined rate of work is \(\frac{1}{23}+\frac{1}{17}\) of the tank per hour.

4. 23 and 17 are both prime, so their LCM is their product: \(23\times17=391\). Thus \(\frac{1}{23}+\frac{1}{17}=\frac{17+23}{391}=\frac{40}{391}\) of the tank per hour.

5. The time to fill the whole tank together is the reciprocal of the combined rate: \(\frac{391}{40}\) hours.

6. Converting to a mixed fraction: \(391=9\times40+31\), so \(\frac{391}{40}=9\frac{31}{40}\) hours.

Cross-check: The combined time, \(9\frac{31}{40}\) hours (9.775 hours), is less than 17 hours, the time the faster pipe alone would take — exactly as expected when a second pipe adds to the work rate. The mixed-fraction conversion also checks out: \(9\times40+31=391\).

Hence, the tank is filled in \(9\frac{31}{40}\) hours when both pipes work together.