P₁ alone can do $\frac{2}{3}$ of the work in 8 days. P₂ alone can do $\frac{7}{12}$ of the work in 14 days. P₃ alone can do $\frac{8}{15}$ of the work in 16 days. In how many days can they complete the work, if all three work together?

$\frac{120}{19}$ days

P1 = \(\frac{2}{3}\)W = 8 days = 12 days,

P2 = \(\frac{7}{12}\)W = 14 days = 24 days,

P3 = \(\frac{8}{15}\)W = 16 days = 30 days,

⇒ Time required by P1 + P2 + P3 to complete the total work = \(\frac{120}{19}\) days.   ..(\(\frac{Work}{Efficiency}\) = Time)

Therefore, all of them working together will complete the work in \(\frac{120}{19}\) days.