B is $${4 \over 3}$$ times as efficient as A. If A can complete $${5 \over 8}$$th of a given task in 15 days, what fraction of the same task would remain incomplete if B works on it independently for 10 days only?

Let efficiency of A = $$3x$$ units/day

=> Efficiency of B = $$\frac{4}{3} \times 3x = 4x$$ units/day

Let Work to be done = 8 units

=> Work done by A in 15 days = $$15 \times 3x = \frac{5}{8} \times 8$$

=> $$45x = 5$$

=> $$x = \frac{5}{45} = \frac{1}{9}$$

Thus, B's 1 day work = $$4 \times \frac{1}{9} = \frac{4}{9}$$ units

Work done by B in 10 days = $$\frac{4}{9} \times 10 = \frac{40}{9}$$ units

=> Work left = $$8 - \frac{40}{9} = \frac{32}{9}$$

$$\therefore$$ Fraction of work left = $$\frac{\frac{32}{9}}{8}$$

= $$\frac{4}{9}$$