Time and Work

Using the formula M1*D1*H1 = M2*D2*H2, we have 12 * 20 * 3 = 18 * 10 * H2. This simplifies to 720 = 180 * H2, so H2 = 4.

Use the formula (M1 * D1 * H1) / W1 = (M2 * D2 * H2) / W2. Here, (20 * 30 * 7) / 1 = (M2 * 40 * 5) / 1. So 4200 = 200 * M2, which means M2 = 21.

Use the formula (M1 * D1 * H1) / W1 = (M2 * D2 * H2) / W2. (25 * 20 * 5) / 2500 = (20 * D2 * 8) / 1200. 2500 / 2500 = 1 = (160 * D2) / 1200. D2 = 1200 / 160 = 120 / 16 = 7.5 = 15/2.

Using the work formula (M1 * D1 * H1) / W1 = (M2 * D2 * H2) / W2, we can substitute the given values: (30 * 24 * 6) / 1500 = (18 * D2 * 8) / 1800. Simplifying this equation gives 2.88 = 0.08 * D2, which yields D2 = 36 days.

Let original number of carpenters be x. Work = x * 20. After 8 left, (x-8) * 40 = work. So 20x = 40(x-8). 20x = 40x - 320. 20x = 320, x = 16.

A's rate = 1/20, B's = 1/10, C's = 1/30. In 5 days, A does 5/20 = 1/4. In next 4 days, A+B do 4(1/20 + 1/10) = 4(3/20) = 12/20 = 3/5. Total done = 1/4 + 3/5 = 17/20. Remaining = 3/20. A+B+C rate = 1/20 + 2/20 + 1.33/20 = 6/60 = 1/10. Time = (3/20) / (1/10) = 1.5 days.

If 60% is done in 6 days, 10% is done in 1 day. The remaining 40% takes 4 days.

A's rate = 1/12, B's rate = 1/16. In 4 days, A+B do 4 * (1/12 + 1/16) = 4 * (7/48) = 7/12. On day 5, B works alone: 1/16. Total in 5 days = 7/12 + 1/16 = 31/48. This pattern continues until work is done.

12 boys = 15 girls, so 1 boy = 1.25 girls. 24 boys and 6 girls is equivalent to 30 + 6 = 36 girls. Since 15 girls take 48 days for 1 unit of work, 36 girls take (15/36) * 48 = 20 days for 1 unit, and 40 days for twice the work.

Using the work formula M1 * D1 = M2 * D2: 39 * 14 = M2 * 6. M2 = (39 * 14) / 6 = 39 * 7 / 3 = 13 * 7 = 91.

Let rates be A+B=1/6, B+C=1/8, C+A=1/12. Adding these: 2(A+B+C) = 1/6 + 1/8 + 1/12 = (4+3+2)/24 = 9/24 = 3/8. So A+B+C = 3/16. A = (A+B+C) - (B+C) = 3/16 - 1/8 = 3/16 - 2/16 = 1/16. Thus, A takes 16 days.

4 men = 10 days, so 1 man = 40 days. 2 men + 3 women = 10 days. 2*(1/40) + 3*(1/W) = 1/10. 1/20 + 3/W = 1/10. 3/W = 1/20. W = 60 days. 3 men + 3 women = 3*(1/40) + 3*(1/60) = 9/120 + 6/120 = 15/120 = 1/8. Time = 8 days.

Let M be the work of a man and W be the work of a woman. From the equations 14(5M + 12W) = 20(7M + 6W), we find 70M + 168W = 140M + 120W, which simplifies to 48W = 70M or M = 24/35 W. Substituting back, the total work is 14(5(24/35)W + 12W) = 14(24/7 + 84/7)W = 14(108/7)W = 216W. Dividing by 18W gives 12 days.