A woman, to do a piece of work takes half the time than a child takes, and a man takes the same time than woman and child takes together. If all three are working together it takes 21 days, then how long would 6 children take to complete the work?

Let child's time = C days. Woman takes C/2 days, so woman's rate = 2/C. Man's time equals woman and child together: 1/(2/C + 1/C) = 2C/3 days, so man's rate = 3/(2C). Combined rate: 2/C + 1/C + 3/(2C) = (4 + 2 + 3)/(2C) = 9/(2C). They take 21 days together, so 9/(2C) = 1/21, giving C = 189/2 = 94.5. Six children: rate = 6/C = 12/189, so time = 189/12 = 15.75 days. However, interpreting 'man takes same time than woman and child takes together' as sum of individual times gives M = C/2 + C = 3C/2. With this interpretation, C = 126, and 6 children take 126/6 = 21 days.