P can finish a work in 19 days. He start doing the work and after 7 days Q joined him. If both of them together completed the remaining work in \(\frac{20}{19}\) days, then in how many days can Q alone finish \(\frac{13}{19}\) of the same work?

\(\frac{5}{4}\) days

Let total work = 19

P can do it in 19 days, therefore,

Efficiency of P = 1 w/d

Work done by P in 7 days = 7

Remaining work = 19 - 7 = 12

Remaining work done by P + Q in \(\frac{20}{19}\) days

Efficiency of P + Q = \(\frac{12}{\frac{20}{19}}\) = \(\frac{57}{5}\)

Efficiency of Q = \(\frac{57}{5}\) - 1 = \(\frac{52}{5}\)

Q will do \(\frac{13}{19}\) work in = 19 × \(\frac{13}{19}\) × \(\frac{5}{52}\) = \(\frac{5}{4}\) days