Description for Correct answer:
Part filled by X in 1st min and Y in the 2nd min

= \( \Large \left(\frac{1}{6}+\frac{1}{7}\right) = \frac{13}{42} \)

Part filled by (X+Y) working alternately in 6 min

= \( \Large \frac{1}{2} \times \frac{13}{42} \times 6 = \frac{13}{14} \)

Therefore, Remaining part = \( \Large 1 - \frac{13}{14}=\frac{1}{14} \)

Now, it is the turn of x, one-sixth part is filled in 1 min. One-fourteenth part is filled in \( \Large \left(6 \times \frac{1}{14}\right) \) min = \( \Large \frac{3}{7} \)min

Therefore, Required time = \( \Large \left(6 + \frac{3}{7}\right) = 6\frac{3}{7} \) min