Time and Work with Days Off

Let daily rates be $x, y, z$. In 6 days, X works 5 days, Y works 5 days $\implies 5x + 5y = 1 \implies x + y = 1/5$. In 8 days, X works 6 days, Z works 7 days $\implies 6x + 7z = 2$. In 12 days, Y works 10 days, Z works 10 days $\implies 10y + 10z = 3 \implies y + z = 3/10$.

Solving gives $x = 1/10, y = 1/10, z = 2/10$. On days 1 and 2, none of them takes a day off, so all three work together. In 2 days: work done = $2(x + y + z) = 2(0.1 + 0.1 + 0.2) = 2(0.4) = 0.8$. Remaining work = 0.2. On day 3, X takes a day off (since X works 3 days then takes off? No, X takes off after 3 days of work, so X works on day 3). None takes off on day 3! All three work on day 3 as well. Rate per day = 0.4. To finish remaining 0.2 work, they need 0.5 days. Total time = $2 + 0.5 = 2.5$ days.