Mathematics

P's work rate is 1/15 per day and Q's is 1/20 per day. Together, they complete (1/15 + 1/20) = 7/60 of the work per day. In 4 days, they complete 4 × 7/60 = 28/60 = 7/15 of the work. Therefore, the fraction of work left is 1 - 7/15 = 8/15. However, since the question asks for the fraction of work left, the correct answer is 8/15, which corresponds to option A. On rechecking, the initial calculation was correct, so the correct answer is A.

Let the work done by one man in one day be M and by one woman be W. From the first condition: 3M + 7W = 1/10 (work per day). From the second: 4M + 6W = 1/8. Solving these equations gives M = 1/40 and W = 1/50. Therefore, 10 women will complete the work in 1/(10*W) = 50 days.

Let the work done by one man in one day be M and by one woman be W. From the first condition: 2M + 12W = 1/4 (since the job is finished in 4 days). From the second: 4M + 6W = 1/5. Solving these equations gives M = 1/60 and W = 1/20. Therefore, one man and one woman together do (1/60 + 1/20) = 1/15 of the job per day, so they complete the job in 15 days. However, since 15 is not an option, rechecking the calculation: 2M + 12W = 1/4 => 2M + 12W = 0.25; 4M + 6W = 1/5 => 4M + 6W = 0.2. Multiply the first equation by 2: 4M + 24W = 0.5; subtract the second equation: (4M + 24W) - (4M + 6W) = 0.5 - 0.2 => 18W = 0.3 => W = 0.3/18 = 1/60. Substitute W into first equation: 2M + 12*(1/60) = 0.25 => 2M + 0.2 = 0.25 => 2M = 0.05 => M = 0.025 = 1/40. So, one man and one woman together do (1/40 + 1/60) = (3/120 + 2/120) = 5/120 = 1/24 of the job per day, so they complete the job in 24 days. Therefore, the correct answer is C (24 days).

If 3 men or 4 women can complete the work in 43 days, then the work done by 1 man in 1 day is 1/(3×43) and by 1 woman is 1/(4×43). The combined daily work of 7 men and 5 women is 7/(3×43) + 5/(4×43) = (28/12 + 15/12)/43 = 43/12/43 = 1/(12/43) = 43/12. Therefore, total days = 1 / (combined daily work) = 12/43 days, which is approximately 14 days.

Let the work done by A, B, and C in one day be a, b, and c respectively. Given: a + b = 1/20, b + c = 1/30, c + a = 1/40. Adding all three: 2(a + b + c) = 1/20 + 1/30 + 1/40 = (6 + 4 + 3)/120 = 13/120. So, a + b + c = 13/240. Subtracting b + c = 1/30 from this, we get a = 13/240 - 1/30 = (13/240) - (8/240) = 5/240 = 1/48. Hence, A alone finishes the work in 48 days.

First, calculate the total distance covered. Distance = Speed × Time = 240 × 5 = 1200 km. Since the same distance must be covered in 5/3 hours, use the formula Speed = Distance ÷ Time. Therefore, Speed = 1200 ÷ (5/3) = 1200 × (3/5) = 720 km/hr. Hence, the correct answer is Option D (720 km/hr). The provided answer key marking Option A (660 km/hr) is incorrect, as the calculation clearly gives 720 km/hr.

P is 30% more efficient than Q, so the efficiency ratio of P to Q is 130:100 or 13:10. P completes the work in 23 days, so P's work rate is 1/23 work per day. Q's work rate is (10/13) times P's rate, which is 10/(13 × 23). Together, their combined work rate is 1/23 + 10/(13 × 23) = (13 + 10)/(13 × 23) = 23/(13 × 23) = 1/13 work per day. Therefore, working together, they complete the work in 13 days. Hence, the correct answer is option D (13 days).

Each child takes twice the time a man takes to finish the work, so each child works at half the rate of a man. Therefore, 8 children have the combined work rate of 4 men (since 8 × 0.5 = 4). Together, 8 children and 12 men have the combined work rate of 16 men (12 + 4 = 16). They complete the work in 9 days, so the total work is equivalent to 16 men × 9 days = 144 man-days. To find how many days 12 men will take to finish the same work, divide the total work by 12 men: 144 man-days ÷ 12 men = 12 days. Hence, 12 men will finish the work in 12 days.

P, Q, and R's rates of work per day are 1/24, 1/6, and 1/12 respectively. Adding these gives (1/24 + 1/6 + 1/12) = (1/24 + 4/24 + 2/24) = 7/24 work per day. Therefore, the total time to complete the work together is 24/7 days, which is approximately 3 3⁄7 days.

The total work is the same in both cases. Work done by 20 women in 16 days equals work done by 16 men in 15 days. Calculating daily work rates and comparing, the ratio of a man's capacity to a woman's capacity is 2:1.