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Suppose Phil starts pump B at time x.
Pump A works for 5 hours, and pump B works for 6 - x hours.
Pump A empties 1/6 of the pool in 1 hour and pump B empties 1/7 of the pool in 1 hour.
So,
(1/6)*5 + (1/7)(6-x) = 1, and, after multiplying both sides by 42 we obtain 35 + 36 - 6x = 42.
Thus, x = 4 + 5/6 and pump B begins work at 4:50 pm.

Therefore, the answer is D.
Answer: D

Let Phil start pump B at x hours after 1 pm
Pump A works for 5 hours
Pump B works for 6 - x hours
Pump A empties 1/6 of the pool per 1 hour
Pump B empties 1/7 of the pool per 1 hour
$$\left(\frac{1}{6}\cdot\ 5\right)+\left(\frac{1}{7}\cdot\left(6-x\right)\right)=1$$
$$\left(\frac{5}{6}\right)+\frac{\left(6-x\right)}{7}=1$$
$$\frac{35+36-6x}{42}=1$$
$$35+36-6x=42$$
$$71-6x=42$$
$$-6x=43-71$$
$$\frac{-6x}{-6}=\frac{-29}{-6}$$
$$x=4\frac{5}{6}$$
$$x=4+\frac{5}{6}\ \min s$$
x = 4 : 50 pm
Phil should start pump B by 4 : 50 pm
Answer = option D