Time-and-work questions measure rates. If A finishes one job in a days, A's rate is 1/a job per day. Rates add when people work together and subtract when an activity undoes completed work. Choosing an artificial total number of work units can remove fractions, but it does not change the logic.
1. Add daily work, not completion times
If A takes 6 days and B takes 12 days, their rates are 1/6 and 1/12 job/day. Together they do 1/4 job/day and finish in 4 days. Averaging 6 and 12 to get 9 days is wrong because completion times are reciprocals of rates. The combined time must be shorter than the faster worker's time when both help positively.
For two workers, the shortcut ab/(a+b) follows from 1/(1/a+1/b). Use it only when both start together, work continuously at constant rates and finish together. A late start, early departure or alternating schedule needs a timeline.
Worked example 1. A can finish in 8 days and B in 12. Take 24 work units. A completes 3 units/day and B 2. Together they complete 5 units/day, so time is 24/5 = 4.8 days.
2. Workers joining or leaving
Break the job into phases. Compute work completed in each known phase, subtract from the total, then divide the remainder by the rate active in the final phase. Using LCM units keeps arithmetic integer until the last division. The rate after someone leaves is not the original combined rate.
Worked example 2. A takes 10 days; B takes 15. They work together for 3 days, then B leaves. Take total work 30 units. Their daily rates are 3 and 2; together they complete 15 units in 3 days. A alone finishes the remaining 15 units in 5 more days.
3. Efficiency and inverse time
For the same job and equal daily hours, higher efficiency means lower completion time. If A is twice as efficient as B, A takes half as long. “20% more efficient” means A's rate is 1.2 times B's rate; A's time is B's time divided by 1.2, or 5/6 of it. It does not mean 20% less time exactly.
If the job amount changes too, keep work = rate × time. For a fixed output per worker-hour, total work is proportional to workers × days × hours/day. A change in worker count, duration and daily hours can be combined through this product, provided all workers share the same efficiency.
Worked example 3. Twelve workers working six hours daily finish a wall in ten days. At equal output per worker-hour, fifteen workers working eight hours daily need (12×6×10)/(15×8)=6 days for the same wall.
4. Alternate days and periodic helpers
In alternate-day problems, find the amount completed in one full repeating cycle. Then determine the number of complete cycles and simulate the remaining day or days in order. A fractional cycle cannot always be treated like a fractional day, because different people may work on different days.
Worked example 4. A does 1/6 of a job on one day and B does 1/12 on the next, repeating. One two-day cycle completes 1/4. Three cycles finish 3/4 in six days. On day seven A adds 1/6, leaving 1/12; B completes that on day eight. The job finishes in eight days.
5. Negative work and wages
A leak or worker undoing work has a negative rate. If an inlet fills 1/4 tank/hour and a leak drains 1/12 tank/hour, net rate is 1/6 tank/hour. The fill time is six hours if both operate throughout. If the leak is closed midway, divide the timeline into phases.
Wages are normally divided in proportion to actual work contributed, not merely time present. Equal time at different efficiencies means unequal shares. If A works five days at 3 units/day and B works five days at 2 units/day, contributions are 15:10=3:2; a ₹1,000 wage pool divides ₹600 and ₹400.
Recall before practice
1. Explain why rates add but completion times do not. 2. Name the steps for a worker-leaves problem. 3. Relate a 20% efficiency rise to completion time for the same job. 4. Explain why alternating-day schedules require whole-cycle reasoning.
Chapter 11 practice — 50 questions
Choose one option for each item. Keep a separate answer list; explanations follow this set.
1. Suresh can finish a work in 6 days working 12 hours a day. In how many days will Suresh finish it working 8 hours a day?
A. 9 days B. 6 days C. 10 days D. 4 days
2. A can do a work in 16 days. B is 60% more efficient than A. In how many days can B alone do the same work?
A. 25⅗ days B. 8 days C. 6⅖ days D. 10 days
3. A can do a work in 21 days and B in 4 days. Working together they receive ₹1,500 for the work. What is A's share?
A. ₹240 B. ₹1,260 C. ₹340 D. ₹750
4. A and B together can complete a work in 42 days. A alone can complete it in 63 days. In how many days can B alone complete the work?
A. 25⅕ days B. 126 days C. 21 days D. 105 days
5. A sum of money is sufficient to pay the wages of Asha for 30 days or Priya for 56 days. For how many days can the same sum pay the wages of Asha and Priya together?
A. 56 days B. 19 23/43 days C. 43 days D. 15 days
6. Rahul can do a work in 15 days and Vikas in 14 days. Working together they receive ₹2,900 for the work. What is Rahul's share?
A. ₹1,500 B. ₹1,300 C. ₹1,400 D. ₹1,450
7. Rahul can do a work in 42 days and Arjun can do the same work in 6 days. In how many days can they complete the work working together?
A. 24 days B. 5¼ days C. 48 days D. 36 days
8. Sanjay and Vikas together can complete a work in 10 hours. Sanjay alone can complete it in 14 hours. In how many hours can Vikas alone complete the work?
A. 5⅚ hours B. 4 hours C. 24 hours D. 35 hours
9. Amit can do a work in 21 days and Rahul in 28 days. Working together they receive ₹420 for the work. What is Amit's share?
A. ₹180 B. ₹340 C. ₹210 D. ₹240
10. A and B together can complete a work in 16 days. A alone can complete it in 48 days. In how many days can B alone complete the work?
A. 64 days B. 12 days C. 24 days D. 32 days
11. Amit and Rohit together can complete a work in 36 days. Amit alone can complete it in 90 days. In how many days can Rohit alone complete the work?
A. 25 5/7 days B. 60 days C. 54 days D. 126 days
12. Deepak and Vikas together can complete a work in 12 hours. Deepak alone can complete it in 48 hours. In how many hours can Vikas alone complete the work?
A. 60 hours B. 36 hours C. 9⅗ hours D. 16 hours
13. A can do a work in 12 days and B in 14 days. They work together for 6 days, after which A leaves. In how many more days will B finish the remaining work?
A. 6/13 days B. 6/7 days C. 1 day D. 7 days
14. A can do a job in 54 days and B can do the same work in 15 days. In how many days can they complete the work working together?
A. 69 days B. 11 17/23 days C. 39 days D. 34½ days
15. Deepak can finish a work in 18 days working 9 hours a day. In how many days will Deepak finish it working 6 hours a day?
A. 27 days B. 12 days C. 21 days D. 18 days
16. A, B and C can do a work alone in 20, 16 and 60 days respectively. A and B start the work together and C joins them after 2 days. In how many days in all is the work completed?
A. 7 23/31 days B. 8 days C. 8 8/9 days D. 6 days
17. Amit alone can do a work in 40 days and Mohan alone in 50 days. Amit starts the work alone and Mohan joins after 1 days. In how many days in all is the work completed?
A. 22 2/9 days B. 22⅔ days C. 49¾ days D. 21⅔ days
18. Arjun and Amit together can complete a work in 64 days. Arjun alone can complete it in 72 days. In how many days can Amit alone complete the work?
A. 576 days B. 33 15/17 days C. 8 days D. 136 days
19. A can complete a work in 40 days. B is 150% more efficient than A. Working together, in how many days will they complete the work?
A. 20 days B. 16 days C. 28 days D. 11 3/7 days
20. Arjun can do a work in 6 days and Sanjay in 12 days. They work together for 2 days, after which Arjun leaves. In how many more days will Sanjay finish the remaining work?
A. 6 days B. 3 days C. 8 days D. 2 days
21. Amit can do a work in 42 days and Arjun in 16 days. They work together for 5 days, after which Amit leaves. In how many days in all was the whole work completed?
A. 14 2/21 days B. 28⅞ days C. 9 2/21 days D. 11 17/29 days
22. Amit can do a work in 15 days and Mohan in 32 days. If both work together for 6 days, what is the ratio of the work done by Amit to that done by Mohan?
A. 33 : 15 B. 32 : 15 C. 47 : 15 D. 15 : 32
23. Ravi can complete a work in 56 days. Amit is 60% more efficient than Ravi. Working together, in how many days will they complete the work?
A. 35 days B. 28 days C. 45½ days D. 21 7/13 days
24. Suresh can do a work in 8 days and Arjun in 27 days. They work together for 3 days, after which Suresh leaves. In how many days in all was the whole work completed?
A. 7⅑ days B. 16⅞ days C. 13⅞ days D. 6 6/35 days
25. Amit can do a work in 6 days and Sanjay in 24 days. With the help of Rahul, all three finish it in 4 days. In how many days can Rahul alone do the work?
A. 26 days B. 12 days C. 24 days D. 4⅘ days
26. A can do a work in 15 days and B in 42 days. They work together for 11 days, after which A leaves. In how many days in all was the whole work completed?
A. 11 1/14 days B. ⅕ days C. 11 1/19 days D. 11⅕ days
27. 20 workers were engaged to dig a canal in 18 days. After 9 days it was decided that the remaining work must be finished in the next 3 days. How many additional workers are required?
A. 10 B. 42 C. 60 D. 40
28. Rohit and Mohan together can complete a work in 14 hours. Rohit alone can complete it in 28 hours. In how many hours can Mohan alone complete the work?
A. 28 hours B. 42 hours C. 14 hours D. 9⅓ hours
29. A can complete a work in 40 days. B takes 20% less time than A to complete the same work. In how many days will they complete it working together?
A. 32 days B. 17 7/9 days C. 18 2/11 days D. 36 days
30. Vikas can do a work in 14 days and Sanjay in 15 days. With the help of Amit they finish it in 5 days and are paid ₹2,100 in all. How much should Amit get?
A. ₹1,450 B. ₹750 C. ₹650 D. ₹700
31. Deepak alone can do a work in 21 days and Amit alone in 24 days. Deepak starts the work alone and Amit joins after 2 days. In how many days in all is the work completed?
A. 23 5/7 days B. 12 2/15 days C. 10 2/15 days D. 11⅕ days
32. Rahul can do a work in 24 days and Mohan in 14 days. With the help of Vikas they finish it in 6 days and are paid ₹560 in all. How much should Vikas get?
A. ₹140 B. ₹380 C. ₹240 D. ₹180
33. A and B together can complete a work in 9 hours. A alone can complete it in 36 hours. In how many hours can B alone complete the work?
A. 27 hours B. 7⅕ hours C. 12 hours D. 45 hours
34. 20 workers were engaged to dig a canal in 30 days. After 14 days it was decided that the remaining work must be finished in the next 2 days. How many additional workers are required?
A. 142 B. 140 C. 138 D. 160
35. Mohan alone can do a work in 18 days and Arjun alone in 40 days. Mohan starts the work alone and Arjun joins after 2 days. In how many days in all is the work completed?
A. 11 1/29 days B. 13 1/29 days C. 12 12/29 days D. 37 5/9 days
36. Sanjay can do a work in 6 days and Mohan in 16 days. With the help of Vikas, all three finish it in 4 days. In how many days can Vikas alone do the work?
A. 12 days B. 18 days C. 48 days D. 5⅓ days
37. Ravi and Arjun can do a work alone in 12, 15 days respectively. They work on alternate days, Ravi working on the first day. In how many days will the work be completed?
A. 13⅓ days B. 6⅔ days C. 13⅕ days D. 13¼ days
38. Rohit can do a work in 15 days and Rahul in 4 days. If both work together for 3 days, what is the ratio of the work done by Rohit to that done by Rahul?
A. 5 : 15 B. 4 : 16 C. 4 : 15 D. 19 : 15
39. A can do a work in 35 days and B in 27 days. With the help of C they finish it in 9 days and are paid ₹2,100 in all. How much should C get?
A. ₹700 B. ₹540 C. ₹1,240 D. ₹860
40. A can do a work in 10 days and B in 15 days. They work together for 4 days, after which A leaves. In how many more days will B finish the remaining work?
A. 3⅓ days B. 2 days C. 5 days D. 9 days
41. Vikas can bind 60 books in 12 hours. Vikas and Suresh together can bind 210 books in 30 hours. How long will Suresh alone take to bind 24 books?
A. 3 hours B. 12 hours C. 24 hours D. 5 hours
42. A, B and C can do a work alone in 40, 42 and 48 days respectively. A and B start the work together and C joins them after 4 days. In how many days in all is the work completed?
A. 14 14/39 days B. 11 5/9 days C. 16 5/9 days D. 15 5/9 days
43. A can finish a work in 12 days working 6 hours a day, and B can finish the same work in 16 days working 10 hours a day. If A works 9 hours a day for 4 days and B works 15 hours a day for 7 days, what is the ratio of the work done by A to that done by B?
A. 17 : 21 B. 21 : 16 C. 12 : 35 D. 16 : 21
44. Rohit, Ravi and Amit can do a work alone in 12, 4, 16 days respectively. They work on alternate days, in the order Rohit, Ravi, Amit, one per day. In how many days will the work be completed?
A. 7 11/19 days B. 9 days C. 2 10/19 days D. 7½ days
45. A can finish a work in 18 days working 8 hours a day, and B can finish the same work in 24 days working 5 hours a day. If A works 10 hours a day for 2 days and B works 6 hours a day for 3 days, what is the ratio of the work done by A to that done by B?
A. 26 : 27 B. 25 : 28 C. 25 : 27 D. 27 : 25
46. One man does as much work as 2 women. If 5 men and 4 women can finish a work in 18 days, in how many days will 3 men and 2 women finish it?
A. 32⅖ days B. 31½ days C. 33 3/7 days D. 10 2/7 days
47. Sanjay, Arjun and Vikas can do a work alone in 20, 5, 15 days respectively. They work on alternate days, in the order Sanjay, Arjun, Vikas, one per day. In how many days will the work be completed?
A. 9¼ days B. 10 days C. 3 3/19 days D. 9 9/19 days
48. Sanjay alone can do a work in 63 days and Arjun alone in 12 days. Sanjay starts the work. After how many days must Arjun join Sanjay so that the work is completed in exactly 21 days?
A. 42 days B. 14 days C. 13 days D. 8 days
49. Deepak, Amit and Sanjay can do a work alone in 18, 6, 14 days respectively. They work on alternate days, in the order Deepak, Amit, Sanjay, one per day. In how many days will the work be completed?
A. 10 8/9 days B. 10 8/21 days C. 10 8/37 days D. 3 15/37 days
50. Ravi can type 20 pages in 5 hours. Ravi and Priya together can type 100 pages in 10 hours. How long will Priya alone take to type 120 pages?
A. 15 hours B. 30 hours C. 12 hours D. 20 hours
Chapter 11 — Answers and explanations
After checking the key, re-solve any miss without looking at the formula.
1. A. Total work in hours = 6 × 12 = 72 hours. At 8 hours a day, days = 72/8 = 9. (Hours per day and days are inversely proportional: fewer hours per day means more days.)
APAR26-11-12 | Change in hours per day | Easy
2. D. Efficiency ratio A : B = 100 : 160. Time is inversely proportional to efficiency, so time ratio = 160 : 100. B's time = 16 × 100/160 = 10 days. (Reducing 16 by 60% gives 6⅖, which is wrong.)
APAR26-11-07 | Efficiency and time alone | Easy
3. A. Wages are shared in the ratio of work done, i.e. of daily rates. LCM(21, 4) = 84: A = 4, B = 21 units/day, ratio 4 : 21. A's share = ₹1,500 × 4/25 = ₹240. (Sharing in the ratio of days 21 : 4 is the trap — the faster worker gets more, not less.)
APAR26-11-05 | Wages of two workers | Easy
4. B. Total work = LCM(63, 42) = 126 units. (A+B) do 126/42 = 3 units per day; A does 126/63 = 2. So B does 3 − 2 = 1 unit per day and needs 126/1 = 126 days. (Subtracting the times, 63 − 42 = 21, is wrong: rates subtract, not times.)
APAR26-11-08 | Together time and one worker known | Easy
5. B. Treat the sum as 840 units (LCM of 30, 56). Daily wages: Asha = 840/30 = 28, Priya = 840/56 = 15. Together they draw 28 + 15 = 43 units/day, so the sum lasts 840/43 = 19 23/43 days. This is exactly the time-and-work formula ab/(a+b).
APAR26-11-01 | Sum of money and wages | Easy
6. C. Wages are shared in the ratio of work done, i.e. of daily rates. LCM(15, 14) = 210: Rahul = 14, Vikas = 15 units/day, ratio 14 : 15. Rahul's share = ₹2,900 × 14/29 = ₹1,400. (Sharing in the ratio of days 15 : 14 is the trap — the faster worker gets more, not less.)
APAR26-11-03 | Wages of two workers | Easy
7. B. LCM method: total work = LCM(42, 6) = 42 units. Rahul does 42/42 = 1 unit per day, Arjun does 42/6 = 7. Together 1 + 7 = 8 units per day, so time = 42/8 = 5¼ days. (Averaging the two times, 24, is the common mistake.)
APAR26-11-11 | Two workers together | Easy
8. D. Total work = LCM(14, 10) = 70 units. (Sanjay+Vikas) do 70/10 = 7 units per hour; Sanjay does 70/14 = 5. So Vikas does 7 − 5 = 2 units per hour and needs 70/2 = 35 hours. (Subtracting the times, 14 − 10 = 4, is wrong: rates subtract, not times.)
APAR26-11-04 | Together time and one worker known | Easy
9. D. Wages are shared in the ratio of work done, i.e. of daily rates. LCM(21, 28) = 84: Amit = 4, Rahul = 3 units/day, ratio 4 : 3. Amit's share = ₹420 × 4/7 = ₹240. (Sharing in the ratio of days 21 : 28 is the trap — the faster worker gets more, not less.)
APAR26-11-02 | Wages of two workers | Easy
10. C. Total work = LCM(48, 16) = 48 units. (A+B) do 48/16 = 3 units per day; A does 48/48 = 1. So B does 3 − 1 = 2 units per day and needs 48/2 = 24 days. (Subtracting the times, 48 − 16 = 32, is wrong: rates subtract, not times.)
APAR26-11-15 | Together time and one worker known | Easy
11. B. Total work = LCM(90, 36) = 180 units. (Amit+Rohit) do 180/36 = 5 units per day; Amit does 180/90 = 2. So Rohit does 5 − 2 = 3 units per day and needs 180/3 = 60 days. (Subtracting the times, 90 − 36 = 54, is wrong: rates subtract, not times.)
APAR26-11-14 | Together time and one worker known | Easy
12. D. Total work = LCM(48, 12) = 48 units. (Deepak+Vikas) do 48/12 = 4 units per hour; Deepak does 48/48 = 1. So Vikas does 4 − 1 = 3 units per hour and needs 48/3 = 16 hours. (Subtracting the times, 48 − 12 = 36, is wrong: rates subtract, not times.)
APAR26-11-13 | Together time and one worker known | Easy
13. C. Total work = LCM(12, 14) = 84 units; A = 7, B = 6 units/day. In 6 days together: 6 × (7 + 6) = 78 units, remaining 84 − 78 = 6 units. B alone finishes 6/6 = 1 days.
APAR26-11-06 | One worker leaves | Easy
14. B. LCM method: total work = LCM(54, 15) = 270 units. A does 270/54 = 5 units per day, B does 270/15 = 18. Together 5 + 18 = 23 units per day, so time = 270/23 = 11 17/23 days. (Averaging the two times, 34½, is the common mistake.)
APAR26-11-10 | Two workers together | Easy
15. A. Total work in hours = 18 × 9 = 162 hours. At 6 hours a day, days = 162/6 = 27. (Hours per day and days are inversely proportional: fewer hours per day means more days.)
APAR26-11-09 | Change in hours per day | Easy
16. B. Total work = LCM(20, 16, 60) = 240 units; rates 12, 15, 4 units/day. First 2 days: 2 × (12 + 15) = 54 units, remaining 186. All three at 31 units/day: 186/31 = 6 days. Total = 2 + 6 = 8 days.
APAR26-11-34 | Third worker joins later | Medium
17. B. Total work = LCM(40, 50) = 200 units; Amit = 5, Mohan = 4 units/day. Amit alone in 1 days: 5 units, remaining 195 units. Together at 9 units/day: 195/9 = 21⅔ days. Total = 1 + 21⅔ = 22⅔ days.
APAR26-11-22 | One worker joins later | Medium
18. A. Total work = LCM(72, 64) = 576 units. (Arjun+Amit) do 576/64 = 9 units per day; Arjun does 576/72 = 8. So Amit does 9 − 8 = 1 unit per day and needs 576/1 = 576 days. (Subtracting the times, 72 − 64 = 8, is wrong: rates subtract, not times.)
APAR26-11-18 | Together time and one worker known | Medium
19. D. Efficiencies A : B = 100 : 250. If A does 100 units/day, total work = 40 × 100 = 4000 units. Together they do 100 + 250 = 350 units/day, so time = 4000/350 = 11 3/7 days. (B alone would take 16 days; averaging 40 and 16 is the trap.)
APAR26-11-36 | Efficiency and time together | Medium
20. A. Total work = LCM(6, 12) = 12 units; Arjun = 2, Sanjay = 1 units/day. In 2 days together: 2 × (2 + 1) = 6 units, remaining 12 − 6 = 6 units. Sanjay alone finishes 6/1 = 6 days.
APAR26-11-32 | One worker leaves | Medium
21. A. Total work = LCM(42, 16) = 336 units; Amit = 8, Arjun = 21 units/day. In 5 days together: 5 × (8 + 21) = 145 units, remaining 336 − 145 = 191 units. Arjun alone finishes 191/21 = 9 2/21 days. Total time = 5 + 9 2/21 = 14 2/21 days.
APAR26-11-35 | One worker leaves | Medium
22. B. In the same 6 days, work done ∝ rate. Rates are 1/15 and 1/32, so the ratio = (1/15) : (1/32) = 32 : 15 = 32 : 15. The number of days worked together cancels. (Writing 15 : 32 inverts the answer.)
APAR26-11-29 | Ratio of work done | Medium
23. D. Efficiencies Ravi : Amit = 100 : 160. If Ravi does 100 units/day, total work = 56 × 100 = 5600 units. Together they do 100 + 160 = 260 units/day, so time = 5600/260 = 21 7/13 days. (Amit alone would take 35 days; averaging 56 and 35 is the trap.)
APAR26-11-33 | Efficiency and time together | Medium
24. B. Total work = LCM(8, 27) = 216 units; Suresh = 27, Arjun = 8 units/day. In 3 days together: 3 × (27 + 8) = 105 units, remaining 216 − 105 = 111 units. Arjun alone finishes 111/8 = 13⅞ days. Total time = 3 + 13⅞ = 16⅞ days.
APAR26-11-28 | One worker leaves | Medium
25. C. Total work = LCM(6, 24, 4) = 24 units. All three: 24/4 = 6 units per day; Amit = 24/6 = 4, Sanjay = 24/24 = 1. So Rahul = 6 − 4 − 1 = 1 unit per day, i.e. 1/Rahul = 1/4 − 1/6 − 1/24. Rahul alone takes 24/1 = 24 days.
APAR26-11-25 | Helper found from combined time | Medium
26. D. Total work = LCM(15, 42) = 210 units; A = 14, B = 5 units/day. In 11 days together: 11 × (14 + 5) = 209 units, remaining 210 − 209 = 1 units. B alone finishes 1/5 = ⅕ days. Total time = 11 + ⅕ = 11⅕ days.
APAR26-11-38 | One worker leaves | Medium
27. D. Total work = 20 × 18 = 360 worker-days. Done in 9 days: 20 × 9 = 180; remaining 180 worker-days must be finished in 3 days, so workers needed = 180/3 = 60. Additional workers = 60 − 20 = 40. (Reporting 60, the total crew, instead of the extra hands is the trap.)
APAR26-11-26 | Extra workers to finish early | Medium
28. A. Total work = LCM(28, 14) = 28 units. (Rohit+Mohan) do 28/14 = 2 units per hour; Rohit does 28/28 = 1. So Mohan does 2 − 1 = 1 unit per hour and needs 28/1 = 28 hours. (Subtracting the times, 28 − 14 = 14, is wrong: rates subtract, not times.)
APAR26-11-39 | Together time and one worker known | Medium
29. B. B's time = 40 × (100 − 20)/100 = 32 days. Together: ab/(a + b) = 40 × 32/(40 + 32) = 1280/72 = 17 7/9 days. ("20% less time" is not "20% more efficient": 20% less time means 25% more efficient.)
APAR26-11-23 | Takes x% less time, together | Medium
30. C. Total work = LCM(14, 15, 5) = 210 units. All three: 210/5 = 42 units/day; Vikas = 15, Sanjay = 14, so Amit = 42 − 15 − 14 = 13 units/day. Wages in ratio 15 : 14 : 13; Amit's share = ₹2,100 × 13/42 = ₹650.
APAR26-11-31 | Helper's share of wages | Medium
31. B. Total work = LCM(21, 24) = 168 units; Deepak = 8, Amit = 7 units/day. Deepak alone in 2 days: 16 units, remaining 152 units. Together at 15 units/day: 152/15 = 10 2/15 days. Total = 2 + 10 2/15 = 12 2/15 days.
APAR26-11-19 | One worker joins later | Medium
32. D. Total work = LCM(24, 14, 6) = 168 units. All three: 168/6 = 28 units/day; Rahul = 7, Mohan = 12, so Vikas = 28 − 7 − 12 = 9 units/day. Wages in ratio 7 : 12 : 9; Vikas's share = ₹560 × 9/28 = ₹180.
APAR26-11-21 | Helper's share of wages | Medium
33. C. Total work = LCM(36, 9) = 36 units. (A+B) do 36/9 = 4 units per hour; A does 36/36 = 1. So B does 4 − 1 = 3 units per hour and needs 36/3 = 12 hours. (Subtracting the times, 36 − 9 = 27, is wrong: rates subtract, not times.)
APAR26-11-24 | Together time and one worker known | Medium
34. B. Total work = 20 × 30 = 600 worker-days. Done in 14 days: 20 × 14 = 280; remaining 320 worker-days must be finished in 2 days, so workers needed = 320/2 = 160. Additional workers = 160 − 20 = 140. (Reporting 160, the total crew, instead of the extra hands is the trap.)
APAR26-11-37 | Extra workers to finish early | Medium
35. B. Total work = LCM(18, 40) = 360 units; Mohan = 20, Arjun = 9 units/day. Mohan alone in 2 days: 40 units, remaining 320 units. Together at 29 units/day: 320/29 = 11 1/29 days. Total = 2 + 11 1/29 = 13 1/29 days.
APAR26-11-27 | One worker joins later | Medium
36. C. Total work = LCM(6, 16, 4) = 48 units. All three: 48/4 = 12 units per day; Sanjay = 48/6 = 8, Mohan = 48/16 = 3. So Vikas = 12 − 8 − 3 = 1 unit per day, i.e. 1/Vikas = 1/4 − 1/6 − 1/16. Vikas alone takes 48/1 = 48 days.
APAR26-11-17 | Helper found from combined time | Medium
37. D. Total work = LCM(12, 15) = 60 units; per day Ravi = 5, Arjun = 4 units. One 2-day cycle does 9 units, so 6 full cycles (12 days) do 54 units, leaving 6. Then Ravi does 5 units (a full day), then Arjun does 1 unit (1/4 of a day). Total = 13¼ days. (Dividing 60 by the cycle rate without tracking whose turn it is gives the wrong fraction.)
APAR26-11-20 | Two workers on alternate days | Medium
38. C. In the same 3 days, work done ∝ rate. Rates are 1/15 and 1/4, so the ratio = (1/15) : (1/4) = 4 : 15 = 4 : 15. The number of days worked together cancels. (Writing 15 : 4 inverts the answer.)
APAR26-11-30 | Ratio of work done | Medium
39. D. Total work = LCM(35, 27, 9) = 945 units. All three: 945/9 = 105 units/day; A = 27, B = 35, so C = 105 − 27 − 35 = 43 units/day. Wages in ratio 27 : 35 : 43; C's share = ₹2,100 × 43/105 = ₹860.
APAR26-11-40 | Helper's share of wages | Medium
40. C. Total work = LCM(10, 15) = 30 units; A = 3, B = 2 units/day. In 4 days together: 4 × (3 + 2) = 20 units, remaining 30 − 20 = 10 units. B alone finishes 10/2 = 5 days.
APAR26-11-16 | One worker leaves | Medium
41. B. Vikas's rate = 60/12 = 5 books/hour. Combined rate = 210/30 = 7 books/hour, so Suresh's rate = 7 − 5 = 2 books/hour. Time for 24 books = 24/2 = 12 hours.
APAR26-11-44 | Work in units | Difficult
42. D. Total work = LCM(40, 42, 48) = 1680 units; rates 42, 40, 35 units/day. First 4 days: 4 × (42 + 40) = 328 units, remaining 1352. All three at 117 units/day: 1352/117 = 11 5/9 days. Total = 4 + 11 5/9 = 15 5/9 days.
APAR26-11-45 | Third worker joins later | Difficult
43. D. Hourly rates: A = 1/(12 × 6) = 1/72 of the work per hour; B = 1/(16 × 10) = 1/160. Work done: A = 4 × 9/72 = 36/72; B = 7 × 15/160 = 105/160. Ratio = 36 × 160 : 105 × 72 = 5760 : 7560 = 16 : 21. (Comparing only hours worked, 36 : 105, ignores the different rates.)
APAR26-11-47 | Ratio of work with different hours per day | Difficult
44. D. Total work = LCM(12, 4, 16) = 48 units; per day Rohit = 4, Ravi = 12, Amit = 3 units. One 3-day cycle does 19 units, so 2 full cycles (6 days) do 38 units, leaving 10. Then Rohit does 4 units (a full day), then Ravi does 6 units (6/12 of a day). Total = 7½ days. (Dividing 48 by the cycle rate without tracking whose turn it is gives the wrong fraction.)
APAR26-11-46 | Three workers in rotation | Difficult
45. C. Hourly rates: A = 1/(18 × 8) = 1/144 of the work per hour; B = 1/(24 × 5) = 1/120. Work done: A = 2 × 10/144 = 20/144; B = 3 × 6/120 = 18/120. Ratio = 20 × 120 : 18 × 144 = 2400 : 2592 = 25 : 27. (Comparing only hours worked, 20 : 18, ignores the different rates.)
APAR26-11-50 | Ratio of work with different hours per day | Difficult
46. B. Convert to women: 5 men + 4 women = 5 × 2 + 4 = 14 women; 3 men + 2 women = 3 × 2 + 2 = 8 women. Total work = 14 × 18 = 252 woman-days, so time = 252/8 = 31½ days.
APAR26-11-43 | Man = k women equivalence | Difficult
47. B. Total work = LCM(20, 5, 15) = 60 units; per day Sanjay = 3, Arjun = 12, Vikas = 4 units. One 3-day cycle does 19 units, so 3 full cycles (9 days) do 57 units, leaving 3. Then Sanjay does 3 units (a full day). Total = 10 days. (Dividing 60 by the cycle rate without tracking whose turn it is gives the wrong fraction.)
APAR26-11-49 | Three workers in rotation | Difficult
48. C. Total work = LCM(63, 12) = 252 units; Sanjay = 4, Arjun = 21 units/day. Sanjay works all 21 days: 4 × 21 = 84 units, leaving 168 units for Arjun. Arjun needs 168/21 = 8 days, so Arjun must join after 21 − 8 = 13 days. (Answering 8, the days Arjun works, instead of the days Arjun waits is the trap.)
APAR26-11-42 | When must the second worker join | Difficult
49. B. Total work = LCM(18, 6, 14) = 126 units; per day Deepak = 7, Amit = 21, Sanjay = 9 units. One 3-day cycle does 37 units, so 3 full cycles (9 days) do 111 units, leaving 15. Then Deepak does 7 units (a full day), then Amit does 8 units (8/21 of a day). Total = 10 8/21 days. (Dividing 126 by the cycle rate without tracking whose turn it is gives the wrong fraction.)
APAR26-11-41 | Three workers in rotation | Difficult
50. D. Ravi's rate = 20/5 = 4 pages/hour. Combined rate = 100/10 = 10 pages/hour, so Priya's rate = 10 − 4 = 6 pages/hour. Time for 120 pages = 120/6 = 20 hours.
APAR26-11-48 | Work in units | Difficult