Time and Work

A completes 1/9 of the work per day, B completes 1/12 per day. In two days (A then B), they complete 1/9 + 1/12 = 7/36 of the work. After 5 such cycles (10 days), they complete 35/36 of the work. On the 11th day, A works and completes the remaining 1/36, which takes 1/4 of a day. Total time = 10 + 1/4 = 42/4 days.

Since A is thrice as good as B, A's work rate is 3 times B's. Let B take x days to finish the work, then A takes x - 60 days. Using rates: A = 1/(x-60), B = 1/x, and A = 3B, so 1/(x-60) = 3/x. Solving gives x = 90 days for B and 30 days for A. Together, they complete work in 1/(1/90 + 1/30) = 22.5 days.

Let A's work rate be 1/X and B's work rate be 1/(X+5). Together, their combined rate is 1/6. So, 1/X + 1/(X+5) = 1/6. Solving this equation gives X = 12.

Let the total work be 24 units (LCM of 8, 12, and 6). P and Q complete 3 units/day (24/8), Q and R complete 2 units/day (24/12), and P, Q, and R complete 4 units/day (24/6). From these, P + Q = 3, Q + R = 2, and P + Q + R = 4 units/day. Subtracting the first from the third gives R = 1 unit/day. Using Q + R = 2, Q = 1 unit/day. Using P + Q = 3, P = 2 units/day. Therefore, P + R = 2 + 1 = 3 units/day, so P and R together will take 24/3 = 8 days. However, this contradicts the initial calculation; rechecking: P + Q = 3, Q + R = 2, P + Q + R = 4. Adding first two: (P + Q) + (Q + R) = 3 + 2 = 5, which equals P + 2Q + R = 5. Subtracting P + Q + R = 4 from this gives Q = 1 unit/day. Then P + Q = 3 implies P = 2 units/day, and Q + R = 2 implies R = 1 unit/day. So P + R = 3 units/day, so days = 24/3 = 8 days. The correct answer is 8 days, which corresponds to option B. Therefore, the original correct_answer 'B' is correct. The explanation above clarifies the calculation.

Given 8 men can dig a pit in 20 days, the total work is 8 men × 20 days = 160 man-days. A man works half as much again as a boy, meaning a man does 1.5 times the work of a boy. Thus, 1 man = 1.5 boys in work rate. For 4 men and 9 boys, convert boys to men equivalent: 9 boys = 9/1.5 = 6 men equivalent. Total men equivalent = 4 + 6 = 10 men. Time taken = total work / total work rate = 160 man-days / 10 men = 16 days. Therefore, the correct answer is 16 days (option B).

Let the total work be 1 unit. Since R alone takes 50 days, R's work rate is 1/50 per day. P can do the work in the same time as Q and R together, so P's rate = Q's rate + R's rate. Given P and Q together take 10 days, their combined rate is 1/10 per day. Using these relations, we find Q's rate = 1/30 per day, so Q alone can do the work in 30 days.

The efficiency ratio of Rashid and Danish is given as 5:8. Efficiency is inversely proportional to the time taken to complete the work. Therefore, the time taken ratio is the inverse, i.e., 8:5. Let the time taken by Rashid be 8x days and by Danish be 5x days. Given that Danish takes 51 days less than Rashid, we have 8x - 5x = 3x = 51 days, which gives x = 17. Hence, Rashid's time to complete the work is 8 * 17 = 136 days. This matches option D, confirming it as the correct answer.

The leak empties the tank in 9 hours, so leak rate = tank capacity / 9. The inlet fills at 10 liters/min = 600 liters/hour. When both operate, tank empties in 16 hours, so net rate = tank capacity / 16 (emptying). Setting net rate = leak rate - inlet rate, we solve for tank capacity and get approximately 12,444 liters.

A's work rate is 1/30 per day and B's work rate is 1/40 per day. Together, their combined rate is 1/30 + 1/40 = (4/120) + (3/120) = 7/120 work per day. Therefore, they can complete the work in 120/7 days.

Bilal completes 1/15 of the work per day, and Jalal completes 1/12 per day. Together, they complete 1/15 + 1/12 = 4/60 + 5/60 = 9/60 = 3/20 of the work per day. In 4 days, they complete 4 × 3/20 = 12/20 = 3/5 of the work. The fraction left is 1 - 3/5 = 2/5. Therefore, the fraction of the work left after 4 days is 2/5, which corresponds to option C.

Since A is thrice as efficient as B, A takes one-third the time B takes. Given A takes 10 days less than B, let B's time be x days. Then A's time is x - 10 days, and A's time is also x/3. So, x - 10 = x/3, solving gives x = 15 days. However, this contradicts the options. Rechecking: x - 10 = x/3 implies 3x - 30 = x, so 2x = 30, x = 15 days. So B takes 15 days, which corresponds to option A. Therefore, the original correct answer A (15 days) is correct. The initial confusion was in explanation wording. The correct answer is A.

Pipe P fills the cistern in 12 minutes, so its rate is 1/12 per minute. Pipe Q fills it in 16 minutes, so its rate is 1/16 per minute. Both pipes open together for t minutes, then only P runs for the remaining (9 - t) minutes. The total work done is (1/12 + 1/16) * t + (1/12) * (9 - t) = 1. Solving this gives t = 3 minutes, so Q should be closed after 3 minutes.

Given that the total wages for 40 women for 30 days is Rs.21600, the daily wage of one woman is Rs.21600 / (40 * 30) = Rs.18. Since a man's daily wage is double that of a woman, a man's daily wage is Rs.36. To find how many men should work for 25 days to earn Rs.14400, use the equation: number of men * 25 days * Rs.36 = Rs.14400. Solving for the number of men gives 14400 / (25 * 36) = 16 men. Therefore, the correct answer is option C (16).

A's work rate is 1/16 per day, B's is 1/12 per day. Together with C, they finish in 4 days, so combined rate is 1/4 per day. C's rate = 1/4 - (1/16 + 1/12) = 1/4 - (3/48 + 4/48) = 1/4 - 7/48 = 12/48 - 7/48 = 5/48 per day. Thus, C alone takes 48/5 = 9.6 days, which corresponds to option B. However, the calculation shows 9.6 days, so option B is correct, but the original correct_answer was B and options are correct. The explanation was missing and difficulty was missing, so I added them. The original answer B is correct.

Let the cost of keeping 1 cow for 1 day be C and 1 goat for 1 day be G. Given 5G = C, so G = C/5. Total cost for 20 cows and 40 goats for 10 days is 460, so 10*(20C + 40G) = 460. Substitute G = C/5: 10*(20C + 40*(C/5)) = 460 => 10*(20C + 8C) = 460 => 10*28C = 460 => 280C = 460 => C = 460/280 = 23/14. Now, cost for 50 cows and 30 goats for 12 days is 12*(50C + 30G) = 12*(50C + 30*(C/5)) = 12*(50C + 6C) = 12*56C = 672C = 672*(23/14) = 672*1.642857 = 1104. Hence, option A is correct.

A completes 2 days of work at a rate of 1/8 per day, so completes 2/8 = 1/4 of the work. Remaining work is 3/4. B completes two-thirds of the remaining work, which is (2/3)*(3/4) = 1/2 of the total work. B's rate is 1/14 per day, so time taken by B is (1/2)/(1/14) = 7 days. Remaining work after B is 1/4. C finishes this remaining 1/4 work at a rate of 1/16 per day, so time taken by C is (1/4)/(1/16) = 4 days. Total time = 2 + 7 + 4 = 13 days. This matches option A, so the correct answer is A (13 days).

The first pipe fills the tank in 20 minutes, so its rate is 1/20 tank per minute. The second pipe fills the tank in 60 minutes, so its rate is 1/60 tank per minute. Both pipes open for 10 minutes fill (1/20 + 1/60) * 10 = (3/60 + 1/60)*10 = (4/60)*10 = 2/3 of the tank. The remaining 1/3 of the tank is filled by the second pipe alone at 1/60 tank per minute, which takes 20 minutes. Total time = 10 + 20 = 30 minutes.

Given: 10 men can complete the work in 7 days, so the work done by 10 men in 1 day is 1/7. Therefore, work done by 1 man in 1 day is (1/7)/10 = 1/70. Similarly, 10 women can complete the work in 14 days, so work done by 10 women in 1 day is 1/14. Therefore, work done by 1 woman in 1 day is (1/14)/10 = 1/140. Now, work done by 5 men in 1 day = 5 × (1/70) = 1/14. Work done by 10 women in 1 day = 10 × (1/140) = 1/14. Total work done by 5 men and 10 women in 1 day = 1/14 + 1/14 = 1/7. Hence, the total number of days required to complete the work = 1 / (1/7) = 7 days. Therefore, the correct answer is 7 days, which corresponds to option A.

Let the work done by P, Q, and R per day be p, q, and r respectively. From the given: p + q = 1/30, q + r = 1/24, r + p = 1/20. Adding all three: 2(p + q + r) = 1/30 + 1/24 + 1/20 = (4 + 5 + 6)/120 = 15/120 = 1/8, so p + q + r = 1/16. After 10 days working together, work done = 10 × 1/16 = 10/16 = 5/8. Remaining work = 3/8. Since Q and R left, only P works now at rate p = (p + q + r) - (q + r) = 1/16 - 1/24 = (3 - 2)/48 = 1/48 per day. Time for P to finish remaining work = (3/8) ÷ (1/48) = 18 days. However, the calculation shows 18 days, but the options have 8 as B and 18 as C. Rechecking: p = (p + q + r) - (q + r) = 1/16 - 1/24 = (3/48 - 2/48) = 1/48. Remaining work = 3/8. Time = (3/8) / (1/48) = (3/8) × 48 = 18 days. So correct answer is C (18). The original answer was correct. The explanation was missing and difficulty was not set; difficulty is medium. Tags added for clarity.

The total work done by 6 men in 2 days is 6 × 2 = 12 man-days, which produces 10 sofas. Therefore, the man-days required to make one sofa is 12 ÷ 10 = 1.2 man-days. For 8 sofas, the total man-days required is 8 × 1.2 = 9.6 man-days. With 8 men working, the number of days required is 9.6 ÷ 8 = 1.2 days. Hence, the correct answer is 1.2 days, which corresponds to option C.

P's work rate is 1/16 per day, Q's is 1/12 per day. Together with R, they complete the job in 4 days, so combined rate is 1/4 per day. R's rate = (1/4) - (1/16 + 1/12) = 1/4 - (3/48 + 4/48) = 1/4 - 7/48 = 12/48 - 7/48 = 5/48. Therefore, R alone takes 48/5 = 9.6 days, which is 9(3/5) days.

The total work can be calculated as 12 men × 8 hours/day × 10 days = 960 man-hours. To complete the same work in 5 days working 12 hours a day, the number of men required is 960 ÷ (12 hours/day × 5 days) = 16 men.

First, find the total work in man-days for the first wall: 15 men × 30 days = 450 man-days to build 500 m. The work per meter is 450 man-days / 500 m = 0.9 man-days per meter. For a 1.2 km (1200 m) wall, total work = 1200 m × 0.9 man-days/m = 1080 man-days. With 30 men working, days required = 1080 man-days / 30 men = 36 days. However, this contradicts the initial calculation, so let's re-check: 15 men × 30 days = 450 man-days for 500 m, so 1 m requires 450/500 = 0.9 man-days. For 1200 m, total man-days = 1200 × 0.9 = 1080. With 30 men, days = 1080/30 = 36 days. So the correct answer is 36 days, which corresponds to option D. The initial correction was wrong; the original correct_answer D is correct.

C takes 6 days to complete half the work, so C's 1 day work = 1/12 of the work. In 4 days, C completes 4 × 1/12 = 1/3 of the work. The remaining work is 2/3. D takes 5 days to complete one-third of the work, so D's 1 day work = 1/15 of the work. To complete the remaining 2/3 of the work, D needs (2/3) ÷ (1/15) = (2/3) × 15 = 10 days. Therefore, D will complete the rest of the work in 10 days, which corresponds to option A.

The problem states that Farjan and Kashif together complete the task in 30 days, so their combined work rate is 1/30 of the task per day. They work together for 20 days, completing 20/30 = 2/3 of the task. The remaining 1/3 of the task is completed by Kashif alone in 25 days, so Kashif's work rate is (1/3)/25 = 1/75 of the task per day. Farjan's work rate is the combined rate minus Kashif's rate: (1/30) - (1/75) = (5/150) - (2/150) = 3/150 = 1/50 of the task per day. Therefore, Farjan alone takes 50 days to complete the entire task. This matches option A, confirming that the correct answer is A (50).