B is 1.5 times as efficient as A. If A can complete $${6 \over 7}$$th of a given task in 12 days, what fraction of the same task would remain incomplete if B works on it independently for 6 days only?

Let efficiency of A = $$2x$$ units/day

=> Efficiency of B = $$1.5 \times 2x = 3x$$ units/day

Let Work to be done = 7 units

=> Work done by A in 12 days = $$12 \times 2x = \frac{6}{7} \times 7$$

=> $$24x = 6$$

=> $$x = \frac{6}{24} = \frac{1}{4}$$

Thus, B's 1 day work = $$3 \times \frac{1}{4} = \frac{3}{4}$$ units

Work done by B in 6 days = $$\frac{3}{4} \times 6 = \frac{9}{2}$$ units

=> Work left = $$7 - \frac{9}{2} = \frac{5}{2}$$

$$\therefore$$ Fraction of work left = $$\frac{\frac{5}{2}}{7}$$

= $$\frac{5}{14}$$