Official Solution:
Samson can fill an empty bathtub with a cold water tap in 6 minutes and 40 seconds or with a hot water tap in 8 minutes. Draining the full bathtub, while the plug is out, takes 13 minutes and 20 seconds. How long will it take Samson to fill the empty bathtub completely with both the cold and hot taps running, while the plug is out?
A. 16 minutes
B. 12 minutes
C. 8.6 minutes
D. 5 minutes
E. 4.75 minutes
The cold water tap can fill the bathtub in \(6\frac{2}{3}=\frac{20}{3}\) minutes, giving a rate of \(\frac{3}{20}\) tubs per minute. Similarly, the hot water tap can fill the bathtub in 8 minutes, giving a rate of \(\frac{1}{8}\) tubs per minute. On the other hand, draining the bathtub takes \(13\frac{1}{3}=\frac{40}{3}\) minutes, giving a rate of \(\frac{3}{40}\) tubs per minute.
Therefore, the net inflow rate when both the cold and hot water taps are running and the drain is open is given by \(\frac{3}{20} + \frac{1}{8}-\frac{3}{40}=\frac{6}{40} + \frac{5}{40}-\frac{3}{40}=\frac{8}{40}=\frac{1}{5}\) tubs per minute.
Since the rate is the reciprocal of time, the time it will take to fill the bathtub completely is therefore 5 minutes.
Answer: D