They work together for 3 days, after which A leaves and C joins B; B and C together finish the remaining work in 3 days. In how many days can C alone…

A's rate = 1/8, B's rate = 1/12 per day. In 3 days together they complete 3(1/8+1/12) = 3(3/24+2/24) = 3×5/24 = 15/24 = 5/8 of the work, leaving 3/8. B and C together finish this 3/8 in 3 days, so their combined daily rate = (3/8)/3 = 1/8. C's rate = 1/8 − 1/12 = 3/24 − 2/24 = 1/24 of the work per day, so C alone takes 24 days. Always convert 'work left' into a rate by dividing by the days taken to clear it, then subtract the known worker's rate.