**(i) **\(\frac{1}{6}\)

If A can do a piece of work in n days, then A can do \(\frac{1}{n}\) of the work in on day

**(ii)** 18

Number of hours A required do a piece of work : 9 hours

Let number of hours B required do a piece of work : X hours

Number of hours required by A and B together to do a piece of work : 6 hours

Work done by A in one hour: \(\frac{1}{9}\)

Work done by B in one hour: \(\frac{1}{x}\)

Work done by A and B together in a hour: \(\frac{1}{6}\)

Work done by A and B together in one hour: \(\frac{1}{9}+\frac{1}{x}=\frac{x\,+\,9}{9x}=\frac{1}{6}\)

∴ \(\frac{x\,+\,9}{9x}=\frac{1}{6}\)

⇒ 6X +54 = 9X

⇒ 3X = 54

⇒ X = \(\frac{54}{3}=18\)

∴ B can do the work 18 hours

**(iii) **48

Number of hours A required do a piece of work : 16 hours

Number of hours B required do a piece of work : 24 hours

Let number of hours C required to do a piece of work : X hours

Number of hours required by A, B and C together to do a piece of work : 8 hours

Work done by A in one hour: \(\frac{1}{16}\)

Work done by B in one hour: \(\frac{1}{24}\)

Work done by C in one hour: \(\frac{1}{x}\)

Work done by A, B and C together in a one hour : \(\frac{1}{8}\)

Work done by A, B and C together in one hour : \(\frac{1}{16}+\frac{1}{24}+\frac{1}{x}=\frac{5}{48}+\frac{1}{x}\)

= \(\frac{5x\,+\,48}{48x}=\frac{1}{8}\)

∴ \(\frac{5x\,+\,48}{48x}=\frac{1}{8}\)

⇒ 40X +384 = 48X

⇒ 8X = 384

⇒ X = \(\frac{384}{8}\) = 48 hours

∴ C can do the work 48 hours.

**(iv)** \(6\frac{2}{3}\)

If A can do a piece of work in n days, then A can do \(\frac{1}{n}\) of the work in on day

∴ \(\frac{20}{3}=6\frac{2}{3}\)