A can do $\frac{2}{5}$ of a work in 6 days and B can do $\frac{2}{3}$ of the same work in 12 days. A and B worked together for 6 days. C alone completed the remaining work in 8 days. A and C, working together, will complete the same work in:

10 days

A = \(\frac{2}{5}\)W = 6 days ⇒ Complete work in 15 days,

B = \(\frac{2}{3}\)W = 12 days ⇒ Complete work in 18 days,

⇒ A + B worked for 6 days = (6 + 5) x 6 = 66 units.  ..(Efficiency × Days = Total work)

⇒ Remaining work = 90 - 66 = 24 units.

⇒ C completed 24 units in 3 days, therefore it's efficiency is = \(\frac{24}{C}\) = 3 days,

⇒ C's efficiency = 3 (Efficiency)

⇒ Time taken by A + C to complete total work = \(\frac{90}{6 + 3}\) = \(\frac{90}{9}\) = 10 days.      ..(\(\frac{Work}{Efficiency}\) = Time)