A particular work can be completed by $6$ men and $6$ women in $24$ days, whereas the same work can be completed by $8$ men and $12$ women in $15$ days. Find the time taken by $4$ men and $6$ women to complete the same work .

Let M and W be daily work of a man and woman. (6M + 6W) * 24 = 1; (8M + 12W) * 15 = 1. Solving these: 144M + 144W = 120M + 180W => 24M = 36W => M = 1.5W. Substituting, (6*1.5W + 6W) * 24 = 1 => 15W * 24 = 1 => 360W = 1. So 1 woman does 1/360 work/day. 4M + 6W = 4(1.5W) + 6W = 12W. Time = 1 / (12 * (1/360)) = 360/12 = 30 days.