A and B can finish a work working on alternate days in 19 days, when A works on the 1st day. Similarly, they can finish the work working on alternate days in 19(5/9) days when B works on the first day. If C working alone can complete the work in 126 days, then in how many days can the work be completed by A, B and C together?

Whether A starts or B starts, Work in 2 days by them will be same. So, work done in 18 days (9×2) will be also same in both cases. Now work done after 18 days will also be same, When A starts the work, after 18 days, A worked for last one day (19th day). When B starts the work, after 18 days, B worked for 19th day and A worked for 5/9th day in the end. So, A × 1 = B × 1 + A × (5/9) A – A × (5/9) = B 4A/9 = B A: B = 9: 4 So let A can do 9 units per day and B can do 4 units per day. So total work when A starts, => (A+B)’s alternate work in 18 days + A’s work in 19th day => (A+B)’s 2 days alternate work × 9 + 9 units => 13 units × 9 + 9 units = 126 units Now C can do this job in 126 days. So, C’s 1 day work = 126/126 = 1 units Hence A, B & C together can do this job in, => 126/(A+B+C) = 126/(9+4+1) = 126/14 = 9 days