1. **Problem Statement:** X, Y, and Z are software experts who work on upgrading software. Each takes a day off after a fixed number of workdays: X after 3 days, Y after 4 days, and Z after 5 days. They start working together after a common day off. Given: - X and Y together complete 1 job in 6 days. - Z and X together complete 2 jobs in 8 days. - Y and Z together complete 3 jobs in 12 days. Find how many days X, Y, and Z together will take to complete 1 job. 2. **Define variables:** Let the daily work rates (jobs per day) of X, Y, and Z be $x$, $y$, and $z$ respectively. 3. **Adjust for days off:** Each expert works for a cycle of days and then takes a day off: - X works 3 days, then 1 day off: cycle length 4 days, works 3/4 of the time. - Y works 4 days, then 1 day off: cycle length 5 days, works 4/5 of the time. - Z works 5 days, then 1 day off: cycle length 6 days, works 5/6 of the time. Effective daily work rates considering days off are: - $x_{eff} = x \times \frac{3}{4}$ - $y_{eff} = y \times \frac{4}{5}$ - $z_{eff} = z \times \frac{5}{6}$ 4. **Use given combined work rates:** - X and Y together complete 1 job in 6 days, so their combined effective rate is $\frac{1}{6}$ jobs/day: $$x_{eff} + y_{eff} = \frac{1}{6}$$ Substitute: $$x \times \frac{3}{4} + y \times \frac{4}{5} = \frac{1}{6}$$ - Z and X together complete 2 jobs in 8 days, so their combined effective rate is $\frac{2}{8} = \frac{1}{4}$ jobs/day: $$z_{eff} + x_{eff} = \frac{1}{4}$$ Substitute: $$z \times \frac{5}{6} + x \times \frac{3}{4} = \frac{1}{4}$$ - Y and Z together complete 3 jobs in 12 days, so their combined effective rate is $\frac{3}{12} = \frac{1}{4}$ jobs/day: $$y_{eff} + z_{eff} = \frac{1}{4}$$ Substitute: $$y \times \frac{4}{5} + z \times \frac{5}{6} = \frac{1}{4}$$ 5. **Rewrite equations:** $$\frac{3}{4}x + \frac{4}{5}y = \frac{1}{6} \quad (1)$$ $$\frac{3}{4}x + \frac{5}{6}z = \frac{1}{4} \quad (2)$$ $$\frac{4}{5}y + \frac{5}{6}z = \frac{1}{4} \quad (3)$$ 6. **Solve equations:** From (1) and (2), subtract (1) from (2): $$\left(\frac{3}{4}x + \frac{5}{6}z\right) - \left(\frac{3}{4}x + \frac{4}{5}y\right) = \frac{1}{4} - \frac{1}{6}$$ $$\frac{5}{6}z - \frac{4}{5}y = \frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}$$ From (3): $$\frac{4}{5}y + \frac{5}{6}z = \frac{1}{4}$$ Add the two equations: $$\left(\frac{5}{6}z - \frac{4}{5}y\right) + \left(\frac{4}{5}y + \frac{5}{6}z\right) = \frac{1}{12} + \frac{1}{4}$$ $$\frac{10}{6}z = \frac{1}{12} + \frac{3}{12} = \frac{4}{12} = \frac{1}{3}$$ $$z = \frac{1}{3} \times \frac{6}{10} = \frac{6}{30} = \frac{1}{5}$$ Substitute $z=\frac{1}{5}$ into (3): $$\frac{4}{5}y + \frac{5}{6} \times \frac{1}{5} = \frac{1}{4}$$ $$\frac{4}{5}y + \frac{1}{6} = \frac{1}{4}$$ $$\frac{4}{5}y = \frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}$$ $$y = \frac{1}{12} \times \frac{5}{4} = \frac{5}{48}$$ Substitute $y=\frac{5}{48}$ into (1): $$\frac{3}{4}x + \frac{4}{5} \times \frac{5}{48} = \frac{1}{6}$$ $$\frac{3}{4}x + \frac{4}{5} \times \frac{5}{48} = \frac{1}{6}$$ $$\frac{3}{4}x + \frac{4}{48} = \frac{1}{6}$$ $$\frac{3}{4}x + \frac{1}{12} = \frac{1}{6}$$ $$\frac{3}{4}x = \frac{1}{6} - \frac{1}{12} = \frac{2}{12} - \frac{1}{12} = \frac{1}{12}$$ $$x = \frac{1}{12} \times \frac{4}{3} = \frac{4}{36} = \frac{1}{9}$$ 7. **Calculate combined effective rate of X, Y, and Z:** $$x_{eff} + y_{eff} + z_{eff} = x \times \frac{3}{4} + y \times \frac{4}{5} + z \times \frac{5}{6}$$ Substitute values: $$= \frac{1}{9} \times \frac{3}{4} + \frac{5}{48} \times \frac{4}{5} + \frac{1}{5} \times \frac{5}{6}$$ $$= \frac{3}{36} + \frac{20}{240} + \frac{5}{30} = \frac{1}{12} + \frac{1}{12} + \frac{1}{6}$$ $$= \frac{1}{12} + \frac{1}{12} + \frac{2}{12} = \frac{4}{12} = \frac{1}{3}$$ 8. **Find total days to complete 1 job together:** Since combined effective rate is $\frac{1}{3}$ jobs/day, time required is: $$\text{Time} = \frac{1}{\text{rate}} = \frac{1}{\frac{1}{3}} = 3 \text{ days}$$ **Final answer:** 3 days